Landau damping

C´edric Villani

Universite´ de Lyon & Institut Henri Poincare,´ 11 rue Pierre et Marie Curie, F-75231 Paris Cedex 05, FRANCE E-mail address: [email protected]

A` MarseilIe, en juillet 2010

Contents

Foreword 7 Chapter 1. Mean field approximation 9 1.1. The Newton equations 9 1.2. Mean field limit 10 1.3. Precised results 14 1.4. Singular potentials 16 Bibliographical notes 18 Chapter 2. Qualitative behavior of the Vlasov equation 21 2.1. Boundary conditions 21 2.2. Structure 22 2.3. Invariants and identities 23 2.4. Equilibria 25 2.5. Speculations 26 Bibliographical notes 28 Chapter 3. Linearized Vlasov equation near homogeneity 31 3.0. Free transport 31 3.1. Linearization 34 3.2. Separation of modes 36 3.3. Mode-by-mode study 38 3.4. The Landau–Penrose stability criterion 43 3.5. Asymptotic behavior of the kinetic distribution 48 3.6. Qualitative recap 50 3.7. Appendix: The Plemelj formula 52 3.8. Appendix: Analyticity and regularity 53 Bibliographical notes 55 Chapter 4. Nonlinear Landau damping 59 4.1. Basic concerns 59 4.2. Backus’s objection 60 4.3. Nonlinear time scale 60 4.4. Elusive bounds 61 4.5. Numerical simulations 62 3 4 CONTENTS

4.6. Theorem 62 4.7. The information cascade 67 4.8. Scheme for attack 68 Bibliographical notes 70 Chapter 5. Gliding analytic regularity 73 5.1. Preliminary analysis 73 5.2. Algebra norms 74 5.3. Gliding regularity 77 5.4. Functional analysis 78 Bibliographical notes 79 Chapter 6. Characteristics in damped forcing 81 6.1. Damped forcing 81 6.2. Deflection 82 Bibliographical notes 84 Chapter 7. Reaction against an oscillating background 85 7.1. Regularity extortion 85 7.2. Solving the reaction equation 86 7.3. Analysis of the kernel K 88 7.4. Analysis of the integral equation 91 7.5. Effect of singular interactions 93 7.6. Large time estimates via exponential moments 95 Bibliographical notes 97 Chapter 8. Newton’s scheme 99 8.0. The classical Newton scheme 100 8.1. Newton scheme for the nonlinear Vlasov equation 101 8.2. Short time estimates 102 8.3. Largetimeestimates 104 8.4. Main result 111 Bibliographical notes 112 Chapter 9. Conclusions 115 Bibliographical notes 117 Bibliography 119 Abstract. This course was taught in the summer of 2010 in the Centre International des Rencontres Math´ematiques as part of a program on mathematical plasma physics related to the ITER project; it constitutes an introduction to the Landau damping phe- nomenon in the linearized and perturbative nonlinear regimes, fol- lowing the recent work [78] by Mouhot & Villani.

5

Foreword

In 1936, Lev Landau devised the basic collisional kinetic model for plasma physics, now commonly called the Landau–Fokker–Planck equation. With this model he was introducing the notion of relaxation in plasma physics: relaxation `ala Boltzmann, by increase of entropy, or equivalently loss of information. In 1946, Landau came back to this field with an astonishing concept: relaxation without entropy increase, with preservation of information. The revolutionary idea that conservative phenomena may exhibit ir- reversible features has been extremely influential, and later led to the concept of violent relaxation. This idea has also been controversial and intriguing, triggering hun- dreds of papers and many discussions. The basic model used by Landau was the linearized Vlasov–Poisson equation, which is only a formal ap- proximation of the Vlasov–Poisson equation. In the present notes I shall present the recent work by Cl´ement Mouhot and myself, extend- ing Landau’s results to the nonlinear Vlasov–Poisson equation in the perturbative regime. Although this extension is still far from handling the mysterious fully nonlinear regime, it already turned out to be rich and tricky, from both the mathematical and the physical points of view. These notes start with basic reminders about classical particle sys- tems and Vlasov equations, assuming no prerequisite from modeling nor physics. Standard notation is used throughout the text, except maybe for the Fourier transforms: if h = h(x, v) is a function on the position-velocity phase space, then h stands for the Fourier transform in the x variable only, while h stands for the Fourier transform in both x and v variables. Precise conventionsb will be given later on. A preliminary version ofe this course was taught in the summer of 2010 in Cotonou, Benin, on the invitation of Wilfrid Gangbo; it is a pleasure to thank the audience for their interest and enthusiasm. The first version of the notes was mostly typed during the nights of a meeting on wave turbulence organized by Christophe Josserand, in the welcoming library of the gorgeous Domaine des Treilles of the Fonda- tion Schlumberger. Then the notes were polished as I was teaching the

7 8 FOREWORD course, on the invitation of Eric´ Sonnendr¨ucker, as part of the Cem- racs 2010 program on plasma physics and mathematics of ITER, in the Centre International des Rencontres Math´ematiques (CIRM), Luminy, near Marseille, France. I hope this text has retained a bit of the magical atmosphere of work and play which was in the air during that summer in Provence. The notes were later repolished and slightly increased on the occasion of a course in Universit´eClaude Bernard (Lyon, France) in 2011, and after the constructive criticisms of an anonymous referee. This foreword is also an opportunity to honor the memory of Naoufel Ben Abdallah, who tragically passed away, only days before the course in CIRM was held. Naoufel was a talented researcher, an energetic colleague, a reliable leader as well as a lively fellow. I cherish the mem- ory of an astonishing hike which we did together, also with his wife Najla and our common friend Jean Dolbeault, in the Haleakala crater on Hawai‘i, back in 1998. These memories of good times will not fade, and neither will the beauty of Naoufel’s contribution to science. CHAPTER 1

Mean field approximation

The two main classes of kinetic equations are the collisional equa- tions of Boltzmann type, modeling short-range interactions, and the mean field equations of Vlasov type, modeling long-range interactions. The distinction between short-range and long-range does not refer to the decay of the microscopic interaction, but to the fact that the rele- vant interaction takes place at distances which are much smaller than, or comparable to, the macroscopic scale; in fact both types of interac- tion may occur simultaneously. Collisional equations are discussed in my survey [104]. In this chapter I will concisely present the archetypal mean field equations.

1.1. The Newton equations The collective interaction of a large population of “particles” arises in a number of physical situations. The basic model consists in the system of Newton equations in Rd (typically d = 3):

(1.1) mi x¨i(t)= Fj→i(t), j X where m is the mass of particle i, x (t) Rd its position at time t,x ¨ (t) i i ∈ i its acceleration, and Fj→i is the force exerted by particle j on particle i. Even if this model does not take into account quantum or relativistic effects, huge theoretical and practical problems remain dependent on its understanding. The masses in (1.1) may differ by many orders of magnitude; actu- ally this disparity of masses plays a key role in the study of the solar system, or the Kolmogorov–Arnold–Moser theory [28], among other things. But it also often happens that the situation where all masses mi are equal is relevant, at least qualitatively. In the sequel, I shall only consider this situation, so mi = m for all i. If the interaction is translation invariant, it is often the case that the force derives from an interaction potential; that is, there is W : Rd R such that → F = W (x y) −∇ − 9 10 1. MEAN FIELD APPROXIMATION is the force exterted at position x by a particle located at position y. This formalism misses important classes of interaction such as magnetic forces, but it will be sufficient for our purposes. Examples 1.1. (a) W (x y) = const. ρ ρ′/ x y is the electrostatic interaction potential between− particles with respective| − | electric charges ρ and ρ′, where x y is the Euclidean distance in R3; (b) W (x y)= const. m m′/ x| −y is| the gravitational interaction potential between− particles− with| respective− | masses m and m′, also in R3; (c) Essentially any potential W arises in some physical problem or the other, and even a smooth (or analytic!) interaction potential W leads to relevant and difficult problems. As an example, let us write the basic equation governing the posi- tions of stars in a galaxy:

xj xi x¨i(t)= mj − 3 , G xj xi Xj6=i | − | where is the gravitational constant. Note that in this example, a star is consideredG as a “particle”! There are similar equations describing the behavior of ions and electrons in a plasma, involving the dielectric constant, mass and electric charges. In the sequel, I will assume that all masses are equal and work in adimensional units, so masses will not explicitly appear in the equa- tions. But now there are as many equations as there are particles, and this means a lot. A galaxy may be made of N 1013 stars, a plasma of N 1020 particles... thus the equations are untractable≃ in practice. Computer≃ simulations, available on Internet, give a flavor of the rich and complex behavior displayed by large particle systems interacting through gravity. It is very difficult to say anything intelligent in front of these complex pictures! This complex behavior is partly due to the fact that the gravita- tional potential is attractive and singular at the origin. But even for a smooth interaction W the large value of N would cause much trouble in the quantitative analysis. The mean field limit will lead to another model, more amenable to mathematical treatment. 1.2. Mean field limit The limit N allows to replace a very large number of simple equations by just→∞ one complicated equation. Although we are trading reassuring ordinary differential equations for dreaded partial differen- tial equations, the result will be more tractable. 1.2. MEAN FIELD LIMIT 11

From the theoretical point of view, the mean field approximation is fundamental: not only because it establishes the basic limit equation, but also because it shows that the qualitative behavior of the system does not depend much on the exact value of the number of particles, and then, in numerical simulations for instance, we can replace trillions of particles by, say, millions or even thousands. It is not a priori obvious how one can let the dimension of the phase space go to infinity. As a first step, let us double variables to convert the second-order Newton equations into a first-order system. So for each position variable xi we introduce the velocity variable vi =x ˙i (time-derivative of the position), so that the whole state of the system d at time t is described by (x1, v1),..., (xN , vN ). Let us write X for the d-dimensional space of positions, which may be Rd, or a subset of Rd, or the d-dimensional torus Td if we are considering periodic data; then the space of velocities will be Rd. Since all particles are identical, we do not really care about the state of each particle individually: it is sufficient to know the state of the system up to permutation of particles. In slightly pedantic terms, we are taking the quotient of the phase space (Xd Rd)N by the permutation × group N , thus obtaining a cloud of undistinguishable points. ThereS is a one-to-one correspondence between such a cloud = (x , v ),..., (x , v ) and the associated empirical measure C { 1 1 N N } 1 N µN = δ , N (xi,vi) i=1 X where δ(x,v) is the Dirac massb in phase space at (x, v). From the physical point of view, the empirical measure counts particles in phase space. Now the empirical measure µN belongs to the space P (Xd Rd), the space of probability measures on the single-particle phase× space. This space is infinite-dimensional,b but it is independent of the number of particles. So the plan is to re-express the Newton equations in terms of the empirical measure, and then pass to the limit as N . For simplicity I shall assume that Xd is either Rd or T→∞d, and that the force derives from an interaction potential W . The following propo- sition establishes the link between the Newton equations and the em- pirical measure equation. Its formulation dispends me from discussing the well-posedness of the Newton equations, which might be tricky if W is not smooth enough (Cauchy–Lipschitz theory would require W to have a locally bounded Hessian) or behaves wildly at infinity (such as W (x)= x4 in R). − 12 1. MEAN FIELD APPROXIMATION

1 d Proposition 1.2. (i) Let W C (X ; R), and for each i let xi = ∈ N −1 xi(t), 0 t T . Then with the notation µ = N δ(xi,x˙i) the following≤ two≤ statements are equivalent: P (1.2) i, x¨ = c W (x b x ) ∀ i − ∇ i − j j X ∂µN (1.3) + v µN + F N (t, x) µN =0, ∂t · ∇x · ∇v where b b b F N (t, x)= c W (x x )= c N W µN . − ∇ − j − ∇ ∗x,v j X
(ii) If W is uniformly continuous and µN convergesb weakly as ∇ 0 N to some measure µ0 and c = c(N) satisfies cN γ 0 → ∞ N → ≥ as N , then up to extraction of a subsequence,b µ converges as t →to ∞ a time-dependent measure µ = µ (dxdv) solving the system →∞ t ∂µ b + v xµ + F (t, x) vµ =0 (1.4) ∂t · ∇ · ∇  F = γ W µ  − ∇ ∗x,v Remark 1.3. Equations (1.3) and (1.4) are to be understood in distributional sense, that is, after integrating on the phase space against a nice test function ϕ(x, v), say smooth and compactly supported. To rewrite these equations in distributional form, note that v µ = (vµ), F (t, x) µ = F (t, x) µ . · ∇x ∇x · · ∇v ∇v · (To be rigorous one should also use a test function in time, bu
t this is not a serious issue and I shall leave it aside.) Remark 1.4. The second formula in (1.4) can be made more ex- plicit as F (t, x)= W (x y) µ (dy dw). − − t ZZ Of course the convolution in the velocity variable is trivial since W does not depend on it; so this is just an integration in velocity space.∇ Remark 1.5. By definition, a sequence of measures µN converges to a measure µ in the weak sense if, for any bounded continuous function ϕ(x, v),

ϕ(x, v) µN (dxdv) ϕ(x, v) µ(dxdv). −−−→N→∞ ZZ ZZ 1.2. MEAN FIELD LIMIT 13

If µN and µ are probability measures, then weak convergence is equiv- alent to convergence in the sense of distributions. Sketch of proof of Proposition 1.2. Let us forget about is- sues of regularity and well-posedness, and focus on the core compu- tations, assuming that xi(t) is a smooth function of t. When we test equation (1.3) against an arbitrary function ϕ = ϕ(x, v) we obtain d 1 1 1 N ϕ(xi, vi) (v xϕ) (F vϕ) =0, dt N −N · ∇ |(xi,vi)−N · ∇ (xi,vi) " i # i i X X X where the time-dependence is implicit; by chain-rule this means 1 ϕ x˙ + ϕ v˙ ϕ v ϕ F N (x ) =0, N ∇x · i ∇v · i − ∇x · i − ∇v · i i X  where ϕ inside the summation is evaluated at (xi, vi). Since vi =x ˙i, this equation reduces to 1 (1.5) v˙ F N (t, x ) ϕ(x , v )=0. N i − i · ∇v i i i X  Now this should hold true for any test function ϕ(x, v). To fix ideas, assume that all (xi, vi) are distinct. Choosing one which takes the form e v near (xi, vi) (with e an arbitrary vector) and which vanishes near · N (xj, vj) for all j = i, we deduce thatv ˙i = F (t, xi). The case when some particles occupy6 the same position in phase space is left as an exercise. (In fact if, say, two particles occupy the same state at some time, then it will be the same for all times, and we can replace them by one single particle.) (This argument is not fully rigorous since it may happen that two distinct particles occupy similar positions in phase space, but that is not a big deal to fix.) Now (1.5) is just a way to rewrite (1.2); the equivalence between (1.2) and (1.3) follows easily. Next we note that W (x xj) = N W µ, where the convo- lution is in both variables∇x and v−. In retrospect,∇ ∗ it is normal that the P force should be expressed in terms of the empiricalb measure, since this is a symmetric expression of the positions of particles. Now let us consider the limit N . Let us fix a finite time- horizon T > 0 and work on the time-interval→ ∞ [ T, T ]. By assumption the initial data µN (0, ) form a tight family; then− from the differential equation satisfied by the· measures µN (t, ) it is not difficult to show N · that µ (t, ) is alsob tight, uniformly in t [ T, T ]. Then, up to extrac- tion of a subsequence,· µN (t, ) will converge∈ − in C([ T, T ]; ′(Xd Rd)) · b − D × b b 14 1. MEAN FIELD APPROXIMATION for any T > 0, to some limit measure µ(t, dx dv). It only remains to pass to the limit in the equation. Being the convolution of a uniformly continuous function with a probability measure, the force field F N = cN W µN is uniformly continuous on [ T, T ] Xd, and will converge− ∇ uniformly∗ as N to γ W µ.− This easily× implies that → ∞ − ∇ ∗ b F N µN Fµ −→ N N in distributional sense, whence v (F µ ) v (Fµ). Similarly, (vµN ) converges to (vµb∇),· and the proof−→ is∇ complete.·  ∇x · ∇x · b The limit equation (1.4) is called the nonlinear Vlasov equation b associated with the interaction potential W . It makes sense just as −1 well for µt(dxdv)= N δ(xi(t),vi(t)) (in which case it reduces to the Newton dynamics) as for µt(dxdv)= f(t, x, v) dxdv, that is, for a con- tinuous distribution of matter.P In fact the nonlinear Vlasov equation is the completion, in the space of measures, of the system of Newton equations. It is customary and physically relevant to restrict to the case of a continuous distribution function, and then focus on the equation satis- fied by f(t, x, v). Since the Lebesgue measure dxdv is transparent to the differential operators x and v, one easily obtains the nonlinear Vlasov equation for the∇ density∇ function f = f(t, x, v): ∂f + v f + F (t, x) f =0 ∂t · ∇x · ∇v (1.6)  F = W ρ, ρ(t, x)= f(t, x, v) dv,  −∇ ∗x Z  where the (x, v)-convolution has been explicitly replaced by a convolu- tion in x and an integration in v. Equation (1.6) is the single most important partial differential equa- tions of mean field systems, and will be the object of study of this course.

1.3. Precised results In Proposition 1.2 it was assumed that W is continuously differen- tiable. If W is smoother then one can prove more precise results of quantitative convergence, involving distances on probability measures, for instance the Wasserstein distances Wp. For the present section, it will be sufficient to know the 1-Wasserstein distance, defined by the 1.3. PRECISED RESULTS 15 formula

W (µ, ν) := sup ψdµ ψ dν; ψ 1 , 1 − k kLip ≤ Z Z  where the supremum is over all 1-Lipschitz functions ψ of (x, v), and it is assumed that µ and ν possess a finite moment of order 1. (If one imposes that ψ is also bounded in supremum norm, one obtains the closely related “bounded Lipschitz” distance, which does not need any moment assumption.) Here is a typical estimate of convergence for the mean-field limit, stated here without proof, going back to Dobrushin:

Proposition 1.6. If µt(dxdv) and νt(dxdv) are two solutions of the nonlinear Vlasov equation with interaction potential W , then for any t R ∈ 2C|t| 2 (1.7) W (µ , ν ) e W (µ , ν ), C = max W ∞ , 1 . 1 t t ≤ 1 0 0 k∇ kL It might not be obvious why this provides a convergence estim
ate in the mean-field limit. To see this, choose µt(dxdv)= f(t, x, v) dxdv N and νt = µt ; then (1.7) controls at time t the distance between the limit mean-field behavior and the Newton equation for N particles, in terms of howb small this distance is at initial time t = 0. If the particles at t = 0 are chosen randomly, then typically the W1 distance at t =0 2C|t| is O(1/√N), so W1(µt, νt)= O(e /√N), which solves the problem. (Note that this estimate requires crazy amounts of particles to get a good precision in large time.) Another type of estimates are large deviation bounds:

2 Proposition 1.7. If W is bounded, f0 = f0(x, v) is given with 2 2 ∇ f (x, v) eβ(|x| +|v| ) dxdv C , (x (0), x˙ (0)), 1 i N, are chosen 0 ≤ 0 i i ≤ ≤ randomly and independently according to f0(x, v) dxdv, (xi(t)) solve RRthe Newton equations (1.2) with c = 1/N, and f(t, x, v) solves the nonlinear Vlasov equation (1.6), then there are K > 0 and N0 > 0 such that for any T 0 there is C = C(T ) such that ≥ (1.8) N N max ε−(2d+3), 1 = ≥ 0 ⇒ 2 P sup W µN , f(t, x,
v) dxdv >ε C 1+ ε−2 e−KNε , 1 t ≤ 0≤t≤T  P  
where stands for probability.b Many refinements are possible: for instance, one can estimate the density error between f(t, x, v) and the empirical measure, after smooth- ing by a peaked convolution kernel; study the evolution of (de)correlations 16 1. MEAN FIELD APPROXIMATION between particles which are initially randomly distributed; show that trajectories of particles in the system of size N are well approximated by trajectories of particles evolving in the limit mean-field force, etc.

1.4. Singular potentials Fine. But eventually, more often than not, the interaction potential is not smooth at all, instead it is rather singular. Then nobody has a clue of why the mean-field limit should be true. The problem might be just technical, but on the contrary it seems very deep. Such is the case in particular for the most important nonlinear Vlasov equations, namely the Vlasov–Poisson equations, where W is the fundamental solution of ∆. In dimension d = 3, writing r = x y , we have ± | − | 1 the Coulomb interaction (repulsive) W = ; • 4πr 1 the Newton interaction (attractive) W = . • −4πr Then the equation F = W ρ becomes F = ∆−1ρ. −∇ ∗ ±∇ It is remarkable that, up to a change of sign in the interaction, the very same equation describes systems of such various scales as a plasma and a galaxy, in which each star counts as one particle! In fact to be more precise, we should slightly change the equation for plasmas, by taking into account the contribution of heavy ions, which is usually considered in the form of a fixed density of positive charges, say ρI (x), and by considering magnetic effects, which in some situations play an important role. Things become much more messy when irreversible phenomena are taken into account, but these phenomena occur only as corrections to the mean-field limit, due to the fact that N is finite. While the mean-field limit for smooth potential has been well- understood for more than three decades, in the case of singular po- tentials the only available results are those obtained a few years ago by Hauray and Jabin: they assume that (a) the interaction is not too singular: essentially W = O(r−s) with 0

The Lions–Perthame theory constructs a unique solution for an initial• datum f on R3 R3 which satisfies, say, i x × v C (1.10) f (x, v) + f(x, v) . | i | |∇ | ≤ (1 + x + v )10 | | | | (The exponent 10 depends on the fact that dimension is 3, and anyway should not be taken seriously.) Besides velocity averaging phenomena, the key insight of the analysis is the propagation of bounds on velocity moments of order greater than 3. Then one can show that the spatial density is uniformly bounded, and the smoothness is propagated too. This elegant theory takes advantage of the dispersion at large positions 18 1. MEAN FIELD APPROXIMATION to control velocity-moments, so it is difficult to adapt to bounded ge- ometries, such as the torus T3; this seems to be a major limitation.

In higher dimension, one expects in general blow-up for the Poisson coupling, at least gravitational; this has been proved in dimension 4 for gravitational interaction if the energy is negative. A discussion for more general singularities and dimensions is still far from sight.

Bibliographical notes Impressive particle simulations of large systems, performed by John Dubinski, can be found online at www.galaxydynamics.org The kinetic theory of plasmas was born in Soviet Union in the thir- ties, when Landau adapted the Boltzmann collision operator to the Coulomb interaction [58] and Vlasov argued that long-range interac- tions should be taken into account by a conceptually simpler mean-field term [108]. The collisional kinetic theory of plasmas is described in a number of physics textbooks [1, 57, 65] and in the mathematical review [104]; see also [2, Sections 1 and 2]. The mean field limit however did not become a mathematical sub- ject until the classical works by Dobrushin [33], Braun & Hepp [21], and Neunzert [81]. Braun & Hepp were also interested in the propaga- tion of chaos and the study of fluctuations; these topics are addressed again in Sznitman’s Saint-Flour lecture notes [100]. Other synthetic sources are the book by Spohn [97] and my incomplete lecture notes on the mean field limit [107], both of which contain a recast of the proof of Proposition 1.6. Quantitative estimates of the mean field limit for simple (stochastic) models and smooth interaction are found in my work [16] joint with Bolley and Guillin; the proof of Proposition 1.7 can be obtained by adapting the estimates therein. The mean-field limit for mildly singular interactions was considered by Hauray and Jabin [44] in a pioneering work that still needs to be digested and simplified by the mathematical community. Early contributions to the Cauchy problem for the Vlasov–Poisson equation, working either in short time, or with weak solutions, or in small dimension, are due to Arsen’ev, Horst, Bardos, Degond, Bena- chour, DiPerna & Lions in the seventies and eighties [6, 7, 9, 13, 29, 49, 50]. The theory reached a more mature stage with the groundbreaking works by Pfaffelmoser [87] and Lions & Perthame [64] at the dawn of the nineties. Pfaffelmoser’s approach was simplified by Schaeffer [95] and is well exposed by Glassey [37]; the adaptation to periodic data BIBLIOGRAPHICAL NOTES 19 was performed by Batt & Rein [11]. The most satisfactory result in this direction is the clever variant by Horst [51], which does not need any compact support assumption, and can be easily transposed to a periodic setting. As for the Lions–Perthame alternative approach, it is reviewed by Bouchut [18]. Blow up in dimension 4 is easy to prove thanks to the virial identity; it is only recently that the blow-up has been studied qualitatively [60].

CHAPTER 2

Qualitative behavior of the Vlasov equation

In the previous chapter we were interested in the derivation and well-posedness of the Vlasov equation ∂f + v f + F (t, x) f =0 ∂t · ∇x · ∇v (2.1)  F = W ρ ρ(t, x)= f(t, x, v) dv.  −∇ ∗x Z But now the emphasis will be different: starting from the Vlasov equa-  tion, we shall enquire about its qualitative behavior. This problem fills up textbooks in physics, and has been the subject of an enormous amount of literature.

2.1. Boundary conditions There is a zoology of boundary conditions for the Vlasov equation. To avoid discussing them, I shall continue to assume that the position space is either Xd = Rd, the whole space, or Xd = Td/Zd, the d- dimensional torus. The latter case deserves some comments. If W is a given potential in Rd, then in the periodic setting, formally W should be replaced by its periodic version W per: W per(x)= W (x k). − Zd kX∈ If W decays fast enough, this is well-defined, but if W has slow decay, like in the case of Poisson interaction, this will not converge! Then the justification requires some argument. In fact, it is clear that for Poisson coupling the potential cannot converge: in the case of the Poisson coupling, the total potential W ρ should formally be equal to ∆−1ρ, which does not make sense since∗ ρ does not have zero mean... ± To get around this problem, we would like to take out the mean of ρ. In the plasma case, one can justify this by going back to the model: indeed, one may argue that the density of ions should be taken into account, that it can be modelled as a uniform background because ions are much heavier and move on longer time scales than electrons, and that the density of ion charges should be equal to the mean density of 21 22 2. QUALITATIVE BEHAVIOR OF THE VLASOV EQUATION electrons because the plasma should be globally neutral. This amounts to replace the potential W ρ by W (ρ ρ ), where ρ = ρ dx. The preceding reasoning∗ is based∗ on the−h existencei ofh twoi different species of particles. But even if there is just one species of particles,R as is the case for gravitational interaction, it is still possible to argue that the mean should be removed. Indeed, in (2.1) W only appears through its gradient, and, whenever c is a constant, W (ρ c)= W ρ W c = W ρ. ∇ ∗ − ∇ ∗ − ∇ ∗ ∇ ∗ Thus, if W decays fast enough at infinity and ρ is periodic,

Rd Td W ρ = W per ρ = W per (ρ ρ ). ∇ ∗ ∇ ∗ ∇ ∗ −h i (I have used the same symbol ρ for a periodic function on Rd and for the function which it induces on Td.) If W does not decay fast enough at infinity, then at least we can write W = limε→0 Wε, where Wε is an approximation decaying fast at infinity (say e−r/ε/(4πr)), then per ± Wε ρ = Wε (ρ ρ ), which in the limit ε 0 converges to ∇W per∗ (ρ ∇ ρ ).∗ Of course−h i this might not be so→ convincing in the ∇absence∗ of a−h cleari discussion of the meaning of the parameter ε, but at least makes sense in some regime and allows to take out the mean ρ from the density in (2.1). This operation is similar to the so-called Jeansh i swindle in astrophysics. Having warned the reader that there is a subtle point here, from now on in the periodic setting I shall always write W ρ for W (ρ ρ ). As a final comment, one may argue against∇ the∗ relevance∇ of∗ peri−hodici boundary conditions, especially in view of the above discussion; but this is still by far the simplest way to have access to a confined geometry, avoiding effects such as dispersion at infinity which completely change the qualitative behavior of the nonlinear Vlasov equation.

2.2. Structure The nonlinear Vlasov equation is a transport equation, and can therefore be solved by the well-known method of characteristics: if f solves the equation, then the measure f(t, x, v) dxdv is the push- forward of the initial measure fi(x, v) dxdv by the flow S0,t =(Xt,Vt) in phase space, solving the characteristic equations

X˙ = V , V˙ = F (t, X ), F = W ρ, t t t t −∇ ∗  (X0,V0)=(x, v).  2.3. INVARIANTS AND IDENTITIES 23

Of course this does not solve the problem “explicitly”, since the force F at time t depends on the whole distribution of particles via the formula F = W ( f dv). −∇ ∗ Recall that the push-forward of a measure µ0 by a map S is defined R −1 by S#µ0[A]= µ0[S (A)]. The resulting equation on the densities gen- erally involves the Jacobian determinant of the flow at time t. However in the present case, the flow St induced by F (t, x) preserves the Liou- ville measure dxdv (that is a consequence of its Hamiltonian nature), so the push-forward equation can be simplified into a pull-back equation for densities. In other words, the solution f(t, x, v) will satisfy

(2.2) f(t, S0,t(x, v)) = f(0, x, v). Thus, to get the distribution function at time t we should invert the map St, in other words solve the characteristics backwards. If St,0 stands for the inverse of S0,t, then (2.2) becomes

(2.3) f(t, x, v)= f(0,St,0(x, v)). Depending on situation, taste and theory, one considers the nonlin- ear Vlasov equation either from the Eulerian point of view (focus on f(t, x, v)), or from the Lagrangian point of view (focus on particle tra- jectories in a force field reconstructed from the particle distribution). This affects not only the theory, but also the numerics, since numer- ical methods may be Eulerian (look at values of f on a grid, say), or Lagrangian (consider particles moving), or semi-Lagrangian (make particles move and interpolate at each step to reconstruct values of f on a grid). Apart from that, equation (2.1) is a limit of Hamiltonian equations (the Newton equations), and actually has a Hamiltonian structure in a certain sense, in relation with optimal transport theory; this link was explored in particular by Ambrosio, Gangbo and Lott. For the moment it is not clear whether this striking structure has physically relevant implications beyond what is already known.

2.3. Invariants and identities In this section I shall review the four main invariances and iden- tities associated with the nonlinear Vlasov equation, assuming that everything is well-defined and being content with formal identities. The nonlinear Vlasov equation preserves the total energy • v 2 1 f(x, v)| | dxdv + W (x y) ρ(x) ρ(y) dxdy =: T + U 2 2 − ZZ ZZ 24 2. QUALITATIVE BEHAVIOR OF THE VLASOV EQUATION is constant in time along solutions. The total energy is the sum of the kinetic energy T and the potential energy U. (The factor 1/2 in the definition of U comes from the fact that we should count unordered pairs of particles. Also, depending on the setting, it may be that the quantity U defined as above is infinite and should be replaced by U ′ = 1/2 W (x y)(ρ(x) ρ )(ρ(y) ρ ) dxdy, which formally is the same as U up− to the constant−h i (1/2)(−h Wi ) ρ 2.) RR h i nonlinear inte- The nonlinear Vlasov equationR preserves all the grals• of the density: often called the Casimirs of the equation, they take the form C(f(x, v)) dxdv, ZZ where C is arbitrary. These millions of conservation laws are imme- diately deduced from (2.3); in other words, they express the fact that the Vlasov equation induces a transport by a measure-preserving (in fact Hamiltonian) flow. In particular, all Lp norms are preserved, the supremum is preserved... and so is the entropy:

S = f log f dxdv. − ZZ The latter property is in sharp contrast with the Boltzmann equation, for which the entropy can only increase in time, unless it is at equilib- rium. Physically speaking, it reflects the preservation of information: whatever information we have about the distribution of particles at initial time, is preserved at later times.

The equation is time-reversible: choose an initial datum fi, let it evolve• by the nonlinear Vlasov equation from time 0 to time T , then reverse velocities (that is replace f(T, x, v) by f(T, x, v)) let it evolve again for an additional time T , reverse velocities again,− and you will be back to the initial datum fi. This again is in contrast with the time-irreversibility of the Boltzmann equation. As a consequence, the nonlinear Vlasov equation does not have any regularizing effect, at least in the usual sense. The last identity is called the virial theorem; it only holds in the• whole space and for specific classes of interaction. The virial1 is defined as = f(x, v) x vdxdv V · ZZ 1This word was made up by Clausius using the latine root for “force”. Clausius advocated for the use of ancient roots for coining new words, and also devised the most successful vocable “entropy”. 2.4. EQUILIBRIA 25 is the time-derivative of the inertia x 2 I = f(x, v) | | dxdv. 2 ZZ If the potential W is even and λ-homogeneous, that is, for any z Rd and α = 0, ∈ 6 W ( z)= W (z), W (αz)= α λ W (z), − | | then one has the virial identity d V =2T λU. dt − The most famous case of application is of course the case of Coulomb/Newton equation, for which λ = 1, so that − d V =2T + U. dt When one takes a time-average and looks over large times, the contribution of the time-derivative is likely to disappear, and we are left with the plausible guess (2.4) 2 T + U =0, h i h i −1 T where u = limT →∞ T 0 u(t) dt. Identity (2.4) suggests some kind of biased,h i but universal partition between the kinetic and potential energies. R

2.4. Equilibria A famous property of the Boltzmann equation is that it only has Gaussian equilibria. In contrast, the Vlasov equation has infinitely many shapes of equilibria. First of all, any distribution f(x, v) = f 0(v) defines a spatially 0 homogeneous equilibrium. Indeed, v xf = 0, and the density ρ0 associated to f 0 is constant, so the corresponding· ∇ force vanishes ( W ρ0 = W ( ρ0) = 0). ∇ The∗ construction∗ ∇ of other classes of equilibria is easy by means of the so-called Jeans theorem: any function of the invariants of the flow is an equilibrium. As the most basic example, let us search for a stationary f in the form of a function of the microscopic energy v 2 E(x, v)= | | + Φ(x), Φ= W ρ, 2 ∗ 26 2. QUALITATIVE BEHAVIOR OF THE VLASOV EQUATION where ρ = f dv. Using the ansatz f(x, v) = f(E), where f is an arbitrary function R R+, we get by chain-rule R → v f Φ f =(f)′(E) v Φ Φ v =0, · ∇x − ∇ · ∇v · ∇ − ∇ · so f is an equilibrium.   Of course this works only if the potential Φ is indeed induced by f, which leads to the compatibility condition v 2 f | | + Φ(y) W (x y) dy dv = Φ(x). 2 − Z   For a given f this is a nonlinear integral equation on the unknown Φ; in the general case it is certainly too hard to solve, but if we are looking for solutions with symmetries, depending on just one parameter, this can often be done in practice. If W is the Coulomb or Newton potential, the integral equation transforms into a differential equation; as a typical situation, consider the three-dimensional gravitational case with radial symmetry, then ρ and Φ are functions of r, and we have after a few computations 0 ρ(r)=4π 2(E Φ(r)) f(E) dE. Φ(r) − Z p This gives ρ as a function of Φ, and then the formulas for spherical Laplace operator applied to radial functions yield 1 d (r2 Φ′(r))=4π ρ(Φ), r2 dr whence f(x, v)= f( v 2/2+Φ(r)) can be reconstructed. Another typical| situation| is the one-dimensional Coulomb interac- tion with periodic data: then the equation is v2 Φ′′(x)= f + Φ(x) dv 1, − 2 − Z   subject to the condition f(v2/2+Φ(x)) dv = 1. Such a solution is called a BGK equilibrium, after Bernstein, Greene and Kruzkal; or BGK wave, to emphasizeR the periodic nature of the solution. Such waves exist as soon as f is smooth and decays fast enough at infinity, and satisfies f(v2/2) dv = 1.

R 2.5. Speculations The general concern by physicists is about the large time asymp- totics, t . Can one somehow draw a picture of the possible quali- tative behavior→∞ of solutions to the nonlinear Vlasov equations? 2.5. SPECULATIONS 27

Usually a first step in the understanding of the large-time behav- ior is the identification of stable structures such as equilibria. In the present case, the abundance of equilibria is a bit disorienting, and we would like to find selection criteria allowing to make predictions in large time. Are equilibria stable? There is a convincing stability criterion for homogeneous equilibria, due to Penrose, which will be studied in Chap- ter 3. But no such thing exists for BGK waves, and nobody has a clue whether these equilibria should be stable or unstable. Having no convincing answer to the previous question, we may turn to an even more difficult question, that is, which equilibria are attrac- tive? Can one witness convergence to equilibrium even in the absence of dissipative features in the equation? Does the Vlasov equation ex- hibit non-entropic relaxation, that is, relaxation without increase of entropy? This has been the object of considerable debate, and sug- gested by numerical experiments on the one hand, observation on the other hand: as pointed out by the astrophysicist Lynden-Bell in the six- ties, galaxies, roughly speaking, seem to be in equilibrium at relevant scales, although the relaxation times associated with entropy produc- tion in galaxies exceed by far the age of the universe. Lynden-Bell argued that there should be a mechanism of violent relaxation, of which nobody has a decent understanding. If the final state is impossible to predict, maybe this problem can be attacked in a statistical way: Lynden-Bell and followers argued that some equilibria, in particular those having high entropy, may be favored by statistical considerations. Maybe there are invariant measures on the space of solutions of the nonlinear Vlasov equation, which can be used to statistically predict the large-time behavior of solutions?? In all this maze of speculations, questions and religions, the only tiny island on which we can stand on our feet is the Landau damp- ing phenomenon: a relaxation property near stable equilibria, which is driven by conservative phenomena. In the sequel I shall describe this phenomenon in great detail; for the moment let me emphasize that besides its theoretical and practical importance by itself, it is the only serious theoretical hint of the possibility of dissipation-free relax- ation in confined systems, without appealing to an extra randomness assumption. 28 2. QUALITATIVE BEHAVIOR OF THE VLASOV EQUATION

Bibliographical notes I am not aware of any good synthetic introductory source for bound- ary conditions of the nonlinear Vlasov equations; but this topic is dis- cussed for instance in the research paper [42]. Boundary conditions for kinetic equations are also evoked in [24, Chapter 8] or [104, Section 1.5]. The Cauchy problem for Vlasov–Poisson in a bounded convex domain is studied in [52]; for nonconvex domains it is expected that serious issues arise about the regularity. The Jeans swindle appears in many textbooks in astrophysics to justify asymptotic expansions when the density is a perturbation of a uniform constant in the whole space, see e.g. [15]. The underlying mathematical meaning of the procedure is neatly explained by Kiessling [55]. The explanations given in Section 2.1 are just an adaptation of this argument to the periodic situation. The Hamiltonian nature of the nonlinear Vlasov equation, in re- lation with optimal transport theory, is discussed informally in my introductory book on optimal transport [105, Section 8.3.2], and more rigorously by Ambrosio & Gangbo [4], and Lott [66, Section 6]. Some of these features are shared by other partial differential equations, in par- ticular the two-dimensional incompressible Euler equation, for which a good concise source is [72]. The similarity between the one-dimensional Vlasov equation and the two-dimensional Euler equation with nonneg- ative vorticity is well-known; physicists have systematically tried to adapt tools and theories from one equation to the other. Early discus- sions of the Hamiltonian nature of these equations, and applications for devising recipes of stability, can be found in [48]. The statistical meaning of the entropy, and its relation to the Boltz- mann formula S = k log W is discussed in many sources; a concise account can be found in my tribute to Boltzmann [106]. Formal properties of the Vlasov equation, including the virial the- orem, are covered in many textbooks such as Binney & Tremaine [15]. This reference also discusses the procedure for constructing inhomoge- neous equilibria. BGK waves were introduced in the seminal paper [14] and have been the object of many speculations in the literature; see [61, 62] for a recent treatment. No BGK wave has been proven to be stable with respect to periodic perturbations (that is, whose period is equal to the period of the wave). The only known related statement is the instability against perturbations with period twice as long [61, 62]. (This holds in dimension 1, but can probably be translated into a multidimensional result.) At least this means that a BGK wave f on T R cannot be × BIBLIOGRAPHICAL NOTES 29 hoped to be stable if f is 1/2-periodic in x. The very recent work [10] presents a new result of the linear damping for some inhomogeneous stationary states. The idea of violent relaxation was introduced in the sixties by Lynden-Bell [67, 68], who at the same time founded the statistical theory of the Vlasov equation. The theory has been pushed by several authors, and also adapted to the two-dimensional incompressible Euler equation [25, 74, 91, 101, 102, 109]. Since it is based on purely heuristic grounds and on just the conservation laws satisfied by the Vlasov equation (not on the equation itself), the statistical theory has been the object of criticism, see e.g. [54]. The construction of invariant measures on infinite-dimensional Hamil- tonian systems has failed for classical equations such as the Vlasov or (two-dimensional, positive vorticity, incompressible) Euler equations [89]; but it was solved for certain dispersive equations, such as the cubic nonlinear Schr¨odinger equations, treated by Bourgain [20]. As far as the Vlasov or Euler equations are concerned, there is no canon- ical choice of what could be a Gibbs measure, but now there might be hope with Sturm’s construction of a fascinating canonical “entropic measure” on the space of probability measures [99], coming from the theory of optimal transport. But for the moment very little is known about Sturm’s measure, and measures drawn according to this measure are not even absolutely continuous.

CHAPTER 3

Linearized Vlasov equation near homogeneity

Vlasov, Landau and other pioneers of the kinetic theory of plasmas discovered a fundamental property: when one linearizes the Vlasov equation around a homogeneous equilibrium, the resulting linear equa- tion is “explicitly” solvable; in a way this is a completely integrable system. This allowed Landau to solve the stability and asymptotic be- havior for the linearized equation — two problems which seem out of reach now for inhomogeneous equilibria.

3.0. Free transport As a preliminary, let us study the properties of free transport, that is, when there is no interaction (W = 0): ∂f (3.1) + v f =0. ∂t · ∇x The properties of this equation differ much in the whole space Rd and in the confined periodic space Td. In the former case, dispersion at infinity dominates the large-time behavior, while in the latter case one observes homogenization phenomena due to phase mixing as illustrated in Fig. 3.1.

v

x

= 10 t = 0 t = 1 t

Figure 3.1. Put an initial disturbance along a line at t = 0. As time goes by, the free transport evolution makes this line twist and homogenize very fast.

Phase mixing occurs for mechanical systems expressed in action- angle variables when the angular velocity genuinely changes with the action variable. In the present case, the angular variable is the position, 31 32 3. LINEARIZED VLASOV EQUATION NEAR HOMOGENEITY so the angular velocity is the plain velocity, which coincides precisely with the action variable.

t = 0 t = 1

t = 100

Figure 3.2. An example of a system which is not mix- ing: for the harmonic oscillator (linearized pendulum) the angular velocity is independent of the action vari- able, so a disturbance in phase space keeps the same shape as time goes by.

The free transport equation can be solved explicitly: if fi is the datum at t = 0, then (3.2) f(t, x, v)= f (x vt, v). i − The simplicity of this formula is deceptive, and the free transport equa- tion displays much trickier behavior than one would imagine at first. To study fine properties of this solution, it is most convenient to use the Fourier transform. Let us introduce the position-velocity Fourier transform

f(k, η)= f(x, v) e−2iπk·x e−2iπη·v dxdv, ZZ where k Zd ise dual to x Td, and η Rd is dual to v Rd. Then (3.2) implies∈ ∈ ∈ ∈

(3.3) f(t, k, η)= f (x vt, v) e−2iπk·x e−2iπη·v dxdv i − ZZ −2iπk·(x+vt) −2iπη·v e = fi(x, v) e e dxdv ZZ = fi(k, η + kt). The last formula has been obtained by just rewriting k (vt) as v (kt) e in the argument of the complex exponential. It is similar· to (3.2), up· to swapping the two variables and changing the direction of time; in fact 3.0. FREE TRANSPORT 33 one may notice that the Fourier transform of the transport equation is a transport equation in Fourier space: ∂f k f =0. ∂t − · ∇η e Anyway, we deduce from (3.3) thate f(t, 0, η)= f (0, η): the zero spatial mode of f is preserved; • i for fixed η and k = 0, fi(k, η + kt) 0 as t , at a • e e 6 −→ → ∞ rate which is (a) determined by the smoothness of fi in v (Riemann– Lebesgue lemma), (b) faster whene k is large. In fact, the relevant time scale for the mode k is k t. | | In particular, if fi is analytic in v then fi decays exponentially fast in η, so the mode k of the solution of the free transport equation will decay like O(e−2πλ|k|t). Also, if f is onlye assumed to be Sobolev regular, say W s,1 in the velocity variable for some s > 0, then the Fourier transform will decay like O( η −s) at large values of η , so the mode of order k will decay like O(( k| t|)−s). | | We can represent this behavior| of| the free transport equation, in Fourier space, as a cascade from low to high velocity modes, the cascade being faster for higher spatial modes. Spatial oscillations generate, in large time, very strong kinetic oscillations.

k t = t t = t (spatial modes) t = t1 2 3

η − (kinetic modes) initial configuration (t = 0)

Figure 3.3. Schematic picture of the evolution of en- ergy by free transport, or perturbation thereof; marks indicate localization of energy in phase space. 34 3. LINEARIZED VLASOV EQUATION NEAR HOMOGENEITY

Remark 3.1. In view of this discussion, the free transport equation appears to be a natural way to convert regularity into a time decay, which can in principle be measured from a physical experiment! Remark 3.2. Even though the free transport equation is reversible, there is a definite behavior as t : f(t, k, η) converges to 0 for all → ±∞ k = 0, and to fi(0, η) for k = 0. This means that 6 e weakly f(t, x, v) fi , e −−−→t→∞ h i where the brackets stand for spatial average:

h (v)= h(x, v) dx. h i Z The convergence holds as long as the initial measure does have a den- sity, that is, fi is well-defined as an integrable function; and it is faster if fi is smooth. Note carefully that the convergence is weak, not strong; in fact the L2 norm of f f does not converge to 0, it remains constant as t . Indeed, −h i →∞ 2 2 2 f f 2 = f dxdv f dv k −h ikL − h i ZZ Z is the difference of two quantities which are separately conserved. Remark 3.3. Why don’t we see such phenomena as recurrence, which are associated with confined mechanical systems? The answer is that as soon as the distribution is spread out and has a density, we do not expect such phenomena because the system truly is infinite- dimensional. Recurrence would occur with a singular distribution func- tion, say Dirac masses, but we ruled out this situation. Of course one may argue that the true physical system is finite, however if the par- ticles are extremely numerous the recurrence times will be enormous, and the behavior is, (hopefully!) accurately described by the continu- ous model.

3.1. Linearization Now let us go back to the Vlasov equation. Let f 0 = f 0(v) be a homogeneous equilibrium. We write f(t, x, v) = f 0(v)+ h(t, x, v), where h 1 in some sense. Since f 0 does not contribute to the force field,k k the ≪ nonlinear Vlasov equation becomes ∂h + v h + F [h] (f 0 + h)=0, ∂t · ∇x · ∇v 3.1. LINEARIZATION 35 where

F [h](t, x)= W (x y) h(t,y,w) dydw = W h. − ∇ − −∇x ∗x,v ZZ

When h is very small we expect the quadratic term F [h] vh to be negligible in front of the linear terms, and obtain the linearized· ∇ Vlasov equation

∂h (3.4) + v h + F [h] f 0 =0. ∂t · ∇x · ∇v The physical interpretation of (3.4) is not so obvious. Assume that we have two species of particles, one that has distribution h and the other one that has distribution f 0, and that the h-particles act on the f 0-particles by forcing, still they are unable to change the distribution f 0 (like you are pushing a wall, to no effect). In this case, we can imagine that the changes in the f 0 density would be compensated by the transmutation of h-particles into f 0-particles, or the reverse. Then the equation for f 0 will be

(3.5) F [h] f 0 = S, · ∇v where S is the source of f 0 particles, and thus the equation for h would be ∂h (3.6) + v h = S. ∂t · ∇x − The combination of (3.5) and (3.6) implies (3.4). Thus, in some sense, equation (3.4) can be interpreted as expressing the reaction exerted by the “wall” f 0 on the particle density. We note that the last term on the right-hand side of (3.4) has the 0 0 form F [h] vf , where F [h] is a function of t and x, and f (v) is a function of·v ∇. This property of separation of variables will be crucial. As a start, it implies the statement below.

Proposition 3.4. If h = h(t, x, v) evolves according to the lin- earized Vlasov equation (3.4), then the function h = h(t, x, v) dx depends only on v and not on t. h i R An equivalent statement is that the linearized Vlasov equation has an infinite number of conservation laws: for any function ψ(v), hψdvdx is a conserved quantity. R 36 3. LINEARIZED VLASOV EQUATION NEAR HOMOGENEITY

Proof of Proposition 3.4. First note that xh = 0 and F [h] = 0, since F [h] is a gradient. So (3.4) implies h∇ i h i

∂ h = v h F [h] f 0 th i −h · ∇x i − · ∇v = v h F [h] f 0 =0. − · h∇x i−h i · ∇v 

3.2. Separation of modes Let us now work on the linearized equation, in the form ∂h (3.7) + v h + F [h] f 0 =0. ∂t · ∇x · ∇v Solving this equation is a beautiful exercice in linear partial differential equations, involving three ingredients (whose order does not matter much): the method of characteristics, the integration in v, and the Fourier transform in x. First step: the method of characteristics. We apply the Duhamel• principle to (3.7), treating it as a perturbation of free trans- port. It is easily checked that the solution of ∂th + v xh = S takes the form · ∇ − t h(t, x, v)= h (x vt, v) S(τ, x v(t τ), v) dτ, i − − − − Z0 where hi(x, v)= h(0, x, v). Second step: Fourier transform. Taking Fourier transform in both• x and v yields (3.8) h(t, k, η)= h (x vt, v) e−2iπk·x e−2iπη·v dxdv i − ZZ t e S(τ, x v(t τ), v) e−2iπk·x e−2iπη·v dxdvdτ − − − ZZ Z0 −2iπk·(x+vt) −2iπη·v (3.9) = hi(x, v) e e dxdv ZZ S(τ, x, v) e−2iπk·x e−2iπk·v(t−τ) e−2iπη·v dxdvdτ − ZZZ t = h (k, η + kt) S τ,k,η + k(t τ) dτ, i − − Z0
e e 3.2. SEPARATION OF MODES 37 where I used the measure-preserving change of variables (x vt, v) (x, v), and the obvious identity k (vs)= v (ks) to absorb the− time-shift→ into a change of arguments in the· Fourier· variables. Now we note that the structure of separated variables in the term S and the properties of Fourier transform imply

S(τ,k,η)= F (τ,k) ^f 0(η) · ∇v =( W ρ1)b(τ,k) ^f 0(η) e b−∇ ∗ · ∇v = 2iπkW (k) ρ1(τ,k) 2iπηf 0(η) − · =4π2 k η W (k) ρ1(τ,k
) f 0(η),
· c b e where ρ1(t, x) = h(t, x, v) dv is the first-order approximation of the c b e spatial density, after the mean density has been taken out. Combining this with (3.8) weR end up with

(3.10) h(t, k, η)= hi(k, η + kt) t e 4π2 W (ke) ρ1(τ,k) f 0(η + k(t τ)) k [η + k(t τ)] dτ. − − · − Z0 c b e Third step: Integrate in v. This amounts to consider the Fourier mode η = 0 in (3.10):

t ρ1(t, k)= h (k,kt) 4π2W (k) ρ1(τ,k) f 0(k(t τ)) k 2 (t τ) dτ. i − − | | − Z0 b To recaste it more synthetically:c b e

t (3.11) ρ1(t, k)= h (k,kt)+ K0(t τ,k) ρ1(τ,k) dτ, i − Z0 where b e b (3.12) K0(t, k)= 4π2 W (k) f 0(kt) k 2t. − | | 1 Z Now appreciate the sheer miracle:c thee Fourier modes ρ (k), k , satisfy a closed equation and evolve in time independently of each other!∈ In a way this expresses a property of complete integrabilityb, which can actually be made more formal. Of course identity (3.12) is interesting only for k = 0; we already 1 6 know that ρ (t, 0) = hi(0, 0) is preserved in time.

b e 38 3. LINEARIZED VLASOV EQUATION NEAR HOMOGENEITY

3.3. Mode-by-mode study If k is given, equation (3.11) is a Volterra equation, which in principle can be solved by Laplace transform. Generally speaking, if we have an equation of the form ϕ = a + K ϕ, that is ∗ t ϕ(t)= a(t)+ K(t τ) ϕ(τ) dτ, − Z0 then it can be changed, via the Laplace transform ∞ (3.13) ϕL(λ)= e2πλt ϕ(t) dt, Z0 into the simple equation ϕL = aL + KL ϕL, whence aL (3.14) ϕL = , 1 KL − which is well-defined at λ R if AL(λ) and KL(λ) are well-defined (for instance if a and K decay∈ exponentially fast and λ is small enough), and (careful!) if KL(λ) = 1. At this point it is useful6 to define the complex Laplace transform: for ξ C, ∈ ∞ ∗ (3.15) ϕL(ξ)= e2πξ t ϕ(t) dt. Z0 It is well-known that the reconstruction of ϕ from its Laplace transform involves integrating ϕL on a well-chosen contour in the complex plane, which has to go out of the real line and should be chosen appropriately. Since Landau, many authors have discussed this tricky issue, by now very classical in plasma physics. However, the reconstruction gives more information than we need: what we want is not the complete description of h, but its time- asymptotics. The following lemma will be enough to achieve this goal: Lemma 3.5. Let K = K(t) be a complex-valued kernel defined for t 0, such that ≥ (i) t 0, K(t) C e−2πλ0t for some C ,λ > 0; ∀ ≥ | | ≤ 0 0 0 (ii) Λ > 0; KL(ξ) =1 for all ξ C with eξ Λ. ∃ 6 ∈ R ≤ Let further a = a(t) be a complex-valued function such that t 0, a(t) α e−2πλt, ∀ ≥ | | ≤ 3.3. MODE-BY-MODE STUDY 39 and let ϕ solve the equation ϕ = a + K ϕ. Then for any λ′ < ∗ min(λ,λ0, Λ),

′ (3.16) t 0, ϕ(t) C α e−2πλ t, ∀ ≥ | | ≤ ′ −1 where C = C(λ,λ , Λ,λ0,C0)(1+ κ ),

(3.17) κ = inf KL(ξ) 1 ; eξ =Λ . | − | R n o Remark 3.6. The fast decay of K at infinity implies that the re- striction of KL(ξ) to the axis eξ = Λ achieves its minimum, therefore κ> 0 under assumptions (i)-(ii).R Further note that by maximum prin- ciple, in the definition of κ it is equivalent to impose the condition eξ = Λ or the apparently weaker condition eξ Λ. R R ≤ Let us express Lemma 3.5 in words: If the kernel K decays ex- ponentially fast and satisfies the stability condition KL = 1 in a half space eξ Λ (see Fig. 3.4), then the solution ϕ decays6 in time at a rate{R which≤ is} limited only by the time-decay of the source, the time-decay of the kernel, and the size of Λ.

ℑm ξ

{KL = 1}

Λ Reξ

Figure 3.4. Λ > 0 is chosen so that the half plane eξ Λ contains no complex root of KL =1 . R ≤ { } 40 3. LINEARIZED VLASOV EQUATION NEAR HOMOGENEITY

Proof of Lemma 3.5. Let us write Φ(t) = e2πλ′t ϕ(t), A(t) = e2πλ′t a(t). The convolution equation ϕ = a + K ϕ becomes ∗ t ′ (3.18) Φ(t)= A(t)+ K(t τ) e2πλ (t−τ) Φ(τ) dτ. − Z0 Taking Laplace transforms yields ΦL(ξ)= AL(ξ)+ KL(λ′ + ξ) ΦL(ξ), whence AL(ξ) (3.19) ΦL(ξ)= , 1 KL(λ′ + ξ) − provided these Laplace transforms are well-defined and KL(λ′ +ξ) = 1. The decay assumption on a immediately implies that AL(ξ) is well-6 defined for eξ < λ λ′; similarly, KL(λ′ + ξ) is well-defined for R′ − ′ eξ<λ0 λ , and does not take the value 1 for eξ < Λ λ . So the Rright hand− side of (3.19) makes perfect sense in aR half-space− eξ<β, where R (3.20) β = min(λ,λ , Λ) λ′ > 0 0 − Let us assume that the left hand side of (3.19) is also well-defined in the same half-space, and obtain the desired conclusion from there. (This estimate will only use the fact that KL(ξ) does not achieve the value 1 on the line eξ = λ′.) To do this, ratherR than inverting the Laplace transform, it is simpler to particularize the identity (3.19) to the imaginary axis, setting ξ = R L ∞ −2iπtω iω, ω . Then Φ (ξ) = Φ(ω) = 0 Φ(t) e dt is the Fourier transform∈ of Φ, where the latter function has been extended by 0 for R t< 0. So (3.19) implies b A(ω) (3.21) Φ(ω)= , ω R. 1 KL(λ′ + iω) ∀ ∈ − b Then, from (3.17)b and Remark 3.6, the denominator in the right-hand side of (3.21) is bounded below by κ, hence

A L2(dω) Φ 2 k k . k kL (dω) ≤ κ b Therefore, by Plancherel’sb identity and the decay assumption on A,

A L2(dt) α Φ L2(dt) k k . k k ≤ κ ≤ κ 4π(λ λ′) − p 3.3. MODE-BY-MODE STUDY 41

Now plug this back in the equation (3.18), to get

2πλ′t Φ L∞(dt) A L∞(dt) + (K e ) Φ k k ≤k k ∗ L∞(dt) 2πλ′t A ∞ + K e 2 Φ 2 ≤k kL (dt) k kL (dt) k kL (dt) C α α + 0 , ≤ 4π(λ λ′) κ 4π(λ λ′) 0 − − whence the desired result. p p Now it is time to come back to identity (3.19) and be more careful about the justification. The logical problem is that Φ does not a priori make sense unless we know that ϕ(t)e2πλ′t belongs to a class where the Fourier transform is well-defined (say L1(dt) for a classicalb definition, or at least in Schwartz class for a definition by duality); but maybe this is an exponentially growing function, in which case Φ might not be well-defined. Of course, the estimate we are looking for establishes an L∞ bound on Φ, but using it to justify the existence ofbΦ would be a circular reasoning. Analytic continuation arguments will be useful to getb out of this trap. First recall that K and A are bounded, so the integral inequality (3.18) implies, via Gronwall’s lemma, the crude bound (3.22) Φ(t) α eCt, C =2πλ′ + C . | | ≤ 0 2πµt L Next, for µ R let Φµ(t) = e Φ(t), so that Φµ(ω) = Φ (µ + iω). ∈ 2πµt 2π(µ+λ′)t Similarly define Aµ(t) = e A(t), Kλ+µ′ (t) = e K(t). Then

Φµ is well-defined for µ< C/(2π), and for suchbµ the identity (3.19) holds with ξ = µ + iω. Equivalently:− b Aµ (3.23) Φµ = . 1 Kλ′+µ − b We would like to take theb analytic continuation right now, but so b far there is no guarantee that Φµ exists as µ approaches 0. So for −δt2/2 δ > 0, let Φµ,δ(t)= e Φµ(t). Then Φµ,δ has a fast decay at infinity, whatever µ R and δ > 0, andb it varies analytically with respect to these parameters.∈ Moreover, replacing multiplication by convolution under the Fourier transform, we have the identity 2 − |ω| Aµ e 2δ Φµ,δ = Φµ γδ = γδ, γδ(ω)= . ∗ 1 Kλ′+µ ! ∗ √2πδ − b This identityb holdsb true only for µ negative enough, but the left hand side and the right hand side areb well-defined and vary analytically with 42 3. LINEARIZED VLASOV EQUATION NEAR HOMOGENEITY respect to their parameters µ, δ and their argument ω (Fourier fre- quency), throughout the region µ < β, where β > 0 is given by (3.20). By analytic continuation, it follows that A Φ0,δ = γδ. 1 Kλ′ ! ∗ −b In particular, since γ b= 1, δ b

R A A L2 Φ0,δ L2 = k k , k k ′ ≤ κ 1 Kλ 2 −b L b −δt2 /2 which provides a bound on Φ(t) e 2 , independent of δ > 0. b L (dt) Letting δ 0 and applyingk the monotonek convergence theorem yields Φ L2(dt→), and we are done.  ∈ Let us apply Lemma 3.5 to (3.11). The kernel K0(t, k) decays as a function of t, exponentially fast if f 0 is analytic, more precisely like −2πλ′|k|t ′ O(e ) for any λ < λ0. (The important remark is that time ap- −2πλi|k|t pears through k t.) Similarly, the source term hi(k,kt) is O(e ) | | 1 if hi is analytic. So in order to ensure the exponential decay of ρ (t, k) −2πλ′|k|t like O(e ), it only remains to check thate there is λL > 0 such that b (3.24) 0 eξ λ k = (K0)L(ξ,k) =1, ≤ R ≤ L| | ⇒ 6 where t ∗ (K0)L(ξ,k)= e2πξ t K0(t, k) dt. Z0 That condition, implicit in Landau’s argument, was introduced by Backus. When it is satisfied, ρ1(t, k) converges to 0 at a rate which is exponential, uniformly for k 1, and more precisely like ′ 1 | | ≥ O(exp( 2πλ k t)). In particular, ρ b(t, ) converges exponentially fast to its mean,− and| | the associated force F [h· ] also converges exponentially fast to 0; this phenomenon is called Landau damping. For mnemonic means, you can figure it in the following way: if you keep pushing on a wall, the wall will not move and you will exhaust yourself. Before going on, note that the conclusion would be different if the position space Xd was the whole space Rd rather than Td: then the spatial mode k would live in Rd rather than Zd (no “infrared cutoff”), and there would be no uniform lower bound for the convergence rate when k becomes small. As a matter of fact, counterexamples by Glassey and Scheffer show that the exponential damping of the force does not hold true in natural norms if Xd = Rd, f 0 is a Gaussian and the 3.4. THE LANDAU–PENROSE STABILITY CRITERION 43 interaction is Coulomb. Numerical computations by Landau suggest that the Landau damping rate in a periodic box of length ℓ decays extremely fast with ℓ, like exp( c/ℓ2). In the sequel, I shall continue− to stick to the case when the position space is Td.

3.4. The Landau–Penrose stability criterion Of course, the previous computation is hardly a solution of the problem, because the stability criterion (3.24) involves the form of the distribution function f 0 in a quite indirect way. Now the problem is to find more explicit stability conditions expressed in terms of f 0. For the moment we shall consider general potentials W . If ξ C satisfies eξ λL k , let us write ξ = (λ + iω) k with λ,ω R∈, λ λ . ThenR ≤ | | | | ∈ ≤ L ∞ (K0)L(ξ,k)= K0(t, k) e2πλ|k|t e−2iπω|k|t dt 0 Z ∞ = 4π2 W (k) f 0(kt) e2πλ|k|t e−2iπω|k|t k 2t dt − | | Z0 ∞ k = 4π2 Wc(k) fe0 u e2πλu e−2iπωu udu. − k Z0 | |  Since f 0(η) = O(e−2πλ0|ηc|), the integrande above is exponentially de- 0 caying as a function of u for λ<λ0. Moreover f (σu) is uniformly continuouse as a function| of| u, uniformly in σ = k/ k . It follows from | | these estimates that (a) by Riemann’s lemma, (Ke0)L(ξ,k) 0 as ω , uniformly in k; (b) (K0)L(ξ,k) is continuous as a function−→ of →∞ λ, uniformly in ω and k. Thus (3.24) will be satisfied for some λL > 0 if and only if it is satisfied for λL = 0. To summarize: under suitable ana- lytic regularity estimates, to check the stability of the linearized Vlasov equation it is sufficient to check that (K0)L(ξ,k) does not achieve the value 1 in the half-space eξ 0 . The next step is to replace{R ≤ the} half-space by the imaginary axis. Let k be given, and for ξ C in a neighborhood of the half-space 0 L∈ eξ 0 let fk(ξ)=(K ) (ξ,k). As noted earlier fk(ξ) goes to 0 as R m ξ≤ ; and it also obviously goes to 0 as eξ . Then, |ℑ |→∞ R → −∞ for Ω > 0 let CΩ be the closed contour in the complex plane made of the line [ iΩ, iΩ] followed by the half-circle Ω e2iπθ, π/2 θ 3π/2. − ≤ ≤ Assume that fk does not achieve the value 1 on the imaginary axis ( e z = 0); then it does not achieve the value 1 on the contour C , R Ω and by Rouch´e’s theorem the number N of roots of (fk(ξ) = 1) in the 44 3. LINEARIZED VLASOV EQUATION NEAR HOMOGENEITY interior of CΩ is given by the integral 1 f ′ (ξ) k dξ. 2π f (ξ) 1 ZCΩ k − As Ω , this approaches →∞ ′ 1 fk(it) dz dt = , Γk = fk(iR). 2π R f (it) 1 z 1 Z k − ZΓk − One can extend Γk into a closed loop by adding the value 0; as ω runs from to + , f (iω) describes a closed path starting from 0 and −∞ ∞ k ending at 0. To make sure that dz/(z 1) = 0, i.e. that Γk does Γk − not circle around 1, it is sufficient to impose that Γ never cross the R k real axis beyond 1, i.e. Γk e z =0 [ , 1). After these preparations,∩ {R we have} obtained⊂ −∞ a stability criterion which is much more tractable: defining ∞ (3.25) Ψ(ω,k)=(K0)b(ω k ,k)= K0(t, k) e−2iπω|k|t dt, | | Z0 check that for any ω R and k Zd 0 , ∈ ∈ \{ } (3.26) m Ψ(ω,k)=0 = e Ψ(ω,k) < 1. ℑ ⇒ R

The next step is to compute a more explicit expression for Ψ(ω,k). From now on we shall assume that W is even: (3.27) z Td, W ( z)= W (z). ∀ ∈ − This natural assumption will imply that W is real-valued. As a start, let us assume d = 1 and k > 0 (so k N). Then we ∈ compute: for any ω R, c ∈ (3.28) ∞ e2iπωkt K0(t, k) dt 0 Z ∞ = lim e−2πλkt e2iπωkt K0(t, k) dt + λ→0 0 Z ∞ = 4π2 W (k) lim f 0(v) e−2iπktv e−2πλkt e2iπωkt k2 tdvdt + − λ→0 0 Z ∞ Z = 4π2 Wc(k) lim f 0(v) e−2iπvt e−2πλt e2iπωt t dv dt. − λ→0+ Z0 Z 0 ′ Then by integrationc by parts, assuming that (f ) is integrable, 1 f 0(v) e−2iπvt t dv = (f 0)′(v) e−2iπvt dv. R 2iπ Z Z 3.4. THE LANDAU–PENROSE STABILITY CRITERION 45

Plugging this back in (3.28), we obtain the expression ∞ 2iπW (k) lim (f 0)′(v) e−2π[λ+i(v−ω)]t dtdv. λ→0+ Z Z0 Next, recall thatc for any λ> 0, ∞ 1 e−2π[λ+i(v−ω)]t dt = ; 2π λ + i(v ω) Z0 − indeed, both sides are holomorphic functions of z = λ + i(v ω) in the half-plane ez > 0 , and they coincide on the real axis −z > 0 , so they have to{R coincide} everywhere. We conclude that { } (f 0)′(v) (3.29) Ψ(ω,k)= W (k) lim dv. λ→0+ v ω iλ Z − − The celebrated Plemeljc formula, recalled in an Appendix, states that 1 1 (3.30) = p.v. + iπ δ in ′(R), x i 0 x 0 D −   where the left-hand side should be understood as the limit, in weak sense, of 1/(z iλ) as λ 0+. The abbreviation p.v. stands for principal value,− that is, simplifying→ the possibly divergent part by using compensations by symmetry when the denominator vanishes; also this notion is recalled in the Appendix. Combining (3.29) and (3.30) we end up with the neat identity for Ψ in (3.26): (f 0)′(v) (3.31) Ψ(ω,k)= W (k) p.v. dv + iπ(f 0)′(ω) . v ω  Z −   Since W is real-valued,c the above formula yields the decomposition of the limit into real and imaginary parts. The problem is to check that the realc part always stays below 1 when the imaginary part vanishes. As soon as (f 0)′(v) = O(1/ v ), we have (f 0)′(v)/(v ω) dv = O(1/ ω ) as ω , so the real| part| in the right-hand side− of (3.31) becomes| | small| |→∞ when ω is large, and we can restrictR the discussion to a bounded interval ω| | Ω. | | ≤ Then the imaginary part, W (k) π(f 0)′(ω), can become small only in the limit k (but then also the real part becomes small) or if → ∞ ω approaches one of the zeroesc of (f 0)′. Since ω varies in a compact set, by continuity it will be sufficient to check the condition only at the zeroes of (f 0)′. In the end, we have obtained the following stability criterion: for any k N, ∈ (f 0)′(v) (3.32) ω R, (f 0)′(ω)=0 = W (k) dv < 1. ∀ ∈ ⇒ v ω Z − c 46 3. LINEARIZED VLASOV EQUATION NEAR HOMOGENEITY

Now if k< 0, we can restart the computation as follows:

∞ e2iπω|k|t K0(t, k) dt = 0 Z ∞ 4π2 W (k) lim f 0(v) e−2iπktv e−2πλ|k|t e2iπω|k|t k 2 tdvdt; − λ→0+ | | Z0 Z then the changec of variables v v brings us back to the previous computation with k replaced by→ −k and f 0(v) replaced by f 0( v). However, it is immediately checked| | that (3.32) is invariant under− re- versal of velocities, that is, if f 0(v) is replaced by f 0( v). Let us summarize what has been achieved. − Proposition 3.7 (Sufficient condition for stability in dimension 1). If W is an even potentiel with W L1(T), and f 0 = f 0(v) is an analytic profile on R such that ∇(f 0)′(∈v) = O(1/ v ), then the Vlasov equation with interaction W , linearized near f 0, is| linearly| stable under analytic perturbations as soon as condition (3.32) is satisfied for all k =0. 6 Let us now examine some examples.

Example 3.8. Consider the Newton interaction, W (k)= 1/ k 2, with a Gaussian distribution − | | c β 2 f 0(v)= ρ0 e−βv /2. r2π Then (3.32) is satisfied if ρ0β < k 2 for all k = 0, that is if β < 1/ρ0: the Gaussian should be spread enough| | to be stable.6 In physics, there is a multiplicative factor in front of the potential, the temperature T = β−1 is typically given andG determines the spreading of the distribution, the density is given, but one can change the size of the periodic box by performing a rescaling in space: the result is that the stability condition is satisfied if and only if L < LJ , where LJ is the so-called Jeans length, πT L = . J ρ0 sG It is widely accepted that this is a typical instability length for the New- tonian Vlasov–Poisson equation, which determines the typical length scale for the inter-galactic separation distance, and thus provides a qualitative answer to the basic question “Why are stars forming clus- ters (galaxies) rather than a uniform background?” 3.4. THE LANDAU–PENROSE STABILITY CRITERION 47

Example 3.9. Consider the Coulomb interaction, W (k)=1/ k 2. If f 0 has only one maximum at the origin, and is nondecreasing| for| v < 0, nonincreasing for v > 0 (for brevity we say thatcf 0 is increas- ing/decreasing), then obviously (f 0)′(v) dv < 0, v Z and (3.32) trivially holds true, independently of the length scale. This is the Landau stability criterion. It works the same for any W such that W 0. ≥ Example 3.10. Still for Coulomb interaction, if f 0 is not increas- ing/decreasing,c then (f 0)′ might have several zeroes, and there is no definite sign for the left-hand side in (3.32). Since all values of W (k) are lying between 0 and 1, (3.32) is implied by the strict inequality (f 0)′(v) c (3.33) ω R, (f 0)′(ω)=0 = dv < 1. ∀ ∈ ⇒ v ω Z − This is the Penrose stability criterion. If f 0 is a small perturbation of an increasing/decreasing distribution, so that it has a slight sec- ondary bump, then the Landau criterion will no longer hold, but the Penrose criterion will still be satisfied, and linear stability will follow. But if the bump is larger, one can have linear instability (bump-on- tail instability, or two-stream instability).

v

Figure 3.5. Bump-on-tail instability: For a given length of the box, there will be linear instability for some large enough secondary bump. Conversely, if a non-monotone velocity distribution is given, there will be instability for some large enough size of the box.

Finally let us generalize this to several dimensions. If k Zd 0 and ξ C, we can use the splitting ∈ \{ } ∈ k k v = r + w, w k, r = v k ⊥ k · | | | | 48 3. LINEARIZED VLASOV EQUATION NEAR HOMOGENEITY and Fubini’s theorem to rewrite ∞ ∗ (K0)L(ξ,k)= 4π2 W (k) k 2 f 0(v) e−2iπkt·v t e2πξ t dv dt − | | Rd Z0 Z ∞ c k ∗ = 4π2W (k) k 2 f 0 r + w dw e−2iπ|k|rtt e2πξ t dr dt, − | | 0 R k r+k⊥ k Z Z Z |k| | |  ! c where k⊥ is the hyperplane orthogonal to k. So everything is expressed in terms of the one-dimensional marginals of f 0. If f is a given function of v Rd, and σ is a unit vector, let us write σ⊥ for the hyperplane orthogonal∈ to σ, and

(3.34) v R fσ(v)= f(w) dw. ∀ ∈ ⊥ Zvσ+σ Then the computation above shows that the multidimensional stabil- ity criterion reduces to the one-dimensional criterion in each direction k/ k . Let us formalize this: | | Definition 3.11 (Penrose’s stability criterion). Let W be an even potential on Td with W L1(Td). We say that f 0 = f 0(v) satisfies the generalized Penrose∇ stability∈ criterion for the interaction potential W if for any k Zd, and any ω R, ∈ ∈ (f 0)′(v) k (f 0)′(ω)=0 = W (k) σ dv < 1, σ = . σ ⇒ v ω k Z − | | Example 3.12. Thec multidimensional generalization of Landau’s stability criterion is that all marginals of f 0 are increasing/decreasing. Example 3.13. If f 0 is radially symmetric and positive, and d 3, then all marginals of f 0 are decreasing functions of v . Indeed,≥ if 2 2 | | ϕ(v) = Rd−1 f( v + w ) dw, then after differentiation and integra- tion by parts we find | | R p ′ 2 2 dw ϕ (v)= (d 3) v f v + w 2 (d 4) − − Rd−1 | | w ≥  Z | | ϕ′(v)= 2π v f( v ) (dp= 3).
 − | |  3.5. Asymptotic behavior of the kinetic distribution Let us assume stability, so that the force F [h] converges to 0 as t , exponentially fast in an analytic setting. What happens to h itself?→ ∞ 3.5. ASYMPTOTIC BEHAVIOR OF THE KINETIC DISTRIBUTION 49

Starting again from

(3.35) h(t, k, η)= hi(k, η + kt) t 2 1 4π eW (k) ρ (eτ,k) f 0(η + k(t τ)) k η + k(t τ) (t τ) dτ − − · − − Z0 we can control the integrand on the right-hand side by the bounds c b e ′ ′ ρ(τ,k) = O(e−2πλ |k|τ ), f 0(η) = O(e−2πλ |η|). | | | | Sacrificing a little bit of the τ-decay of ρ to ensure the convergence of the τ-integral,b using η + kt η + ke|(t| τ) + kτ , and assuming | |≤| − | | | k W (k)= O(1) (which is true if W Lb1), we end up with | | ∇ ∈ t 2 1 2 −2πλ′′|η+kt| 4πcW (k) ρ (τ,k) f 0(η + k(t τ)) k (t τ) dτ = O e , 0 − | | − Z   where λ′′ is arbitrarily close to λ′. Plugging this back in (3.35) implies c b e ′′ (3.36) h(t, k, η) h (k, η + kt) C e−2πλ |η+kt|. − i ≤ It is not difficult to show that the bound (3.36) is qualitatively optimal; it is interesting e only for ke = 0, since we already know h(t, 0, η) = 6 hi(0, η). Let us analyze (3.36) as time becomes large. First, fore each fixed e(k, η) we have h(t, k, η) 0 exponentially fast, in particular −→ t→∞ h(t, ) hi , e · −−−→weakly h i and the speed of convergence is determined by the regularity in velocity space: exponential convergence for analytic data, inverse polynomial for Sobolev data, etc. However, for each t one can find (k, η) such that h(t, k, η) is O(1) (not small!). In other words, the decay of Fourier modes| is not uniform| , and the convergence is not strong. In fact, the spatiale mode k of h(t, ) undergoes oscillations around the kinetic frequency η kt in the· velocity variable as t ; so at time t, the typical oscillation≃ − scale in the velocity variable→ is ∞O(1/ k t) for the mode k. How much high (spatial) frequency modes affect| the| whole distribution h depends on the respective strength of these modes, that is, on the regularity in the x variable; but in any case the kinetic distribution h will exhibit fast velocity oscillations at scales O(1/t) as t goes by. The problem only arises in the velocity variable: it is an easy exercice to check that the smoothness in the position variable is essentially preserved. However, if one considers h along trajectories of free transport, the smoothness is restored: (3.36) shows that h(t, k, η kt) is bounded like − e 50 3. LINEARIZED VLASOV EQUATION NEAR HOMOGENEITY

O(e−2πλ′′|η|), so we do not see oscillations in the velocity variable any longer. Let us call this the gliding regularity: if we change the focus in time to concentrate on modes η kt in Fourier space, we do see a good decay. Equivalently, if we look≃ − at h(x + vt, v), what we see is uniformly smooth as t . Here is an alternative→∞ way to consider this procedure. As t , the force field vanishes, so the linearized Vlasov equation is asymptotic→ ∞ to the free transport evolution. Now the idea is to let the distribution evolve according to the linearized Vlasov for time t, then apply the free evolution backwards from time t to initial time, and study the result. Comparison of the perturbed evolution to the free evolution is very common in classical and quantum mechanics, where it is used to define, in various asymptotic regimes, the so-called interaction representations, wave transforms, and scattering operators.

3.6. Qualitative recap Let me reformulate and summarize what we learnt in this section. I shall start with a precise mathematical statement. Theorem 3.14. Let f 0 = f 0(v) be an analytic homogeneous equi- ^ librium, with f 0(η) = O(e−2πλ0|η|), and let W be an even interaction potential such| that | W L1(Td). Let K0 be defined in (3.12); as- ∇ ∈ 0 L sume that there is λL > 0 such that the Laplace transform (K ) (ξ,k) of K0(t, k) stays away from the value 1 when 0 eξ <λ k . Let ≤ R L| | hi = hi(x, v) be an analytic initial perturbation such that hi(k, η) = O(e−2πλ|η|). Then if h solves the linearized Vlasov equation (3.4) with initial datum hi, one has exponential decay of the force field:e for any k =0, 6 ′ F [h](t, k)= O(e−2πλ |k|t), ′ for any λ < min(λ0,λL,λ). Moreover, Penrose’s stability condition (Definition 3.11) guaranteesd the existence of such a λL > 0. Remark 3.15. Following Landau, physics textbooks usually care 0 only on λL and forget about λ0, λ; so they implicitly assume that f and hi are entire functions (then one can choose λ0 and λL arbitrarily large). But in general one should not forget that the damping rate does 0 depend on the analytic regularity of f and hi. Beyond Theorem 3.14, one can argue that the four key ingredients leading to the decay of the force field are the confinement ensured by the torus; • the mixing property of the geodesic flow (x, v) (x + vt, v); • → 3.6. QUALITATIVE RECAP 51

the Riemann–Lebesgue principle converting smoothness into de- cay• in Fourier space; a stability condition coming fom the study of the Volterra equa- tion,• in this case the generalized Penrose criterion.

The first two ingredients are important: as I already mentioned, there are counterexamples showing that decay does not hold in the whole space, and it is rather well-known from experiments that damp- ing may cease when the flow ceases to be mixing, so that for instance trapped trajectories appear. As for the third ingredient, it is subject to debate, since there are many points of view around as to why damping holds (wave-particle interaction, etc.), but in these notes I will advocate the Riemann–Lebesgue point of view as natural and robust. Now as far as the regularity of h is concerned, one should keep in mind that the regularity of h deteriorates in the velocity variable, as it oscillates• faster and faster in v as time increases; there is a cascade in Fourier space from low to high kinetic modes, which• on the mean is faster for higher position modes — it is like a shear flow in Fourier space; the distribution function evaluated along trajectories of the free flow,• h(t, x + vt, v), remains very smooth, uniformly in time (gliding regularity).

While the regularity of h deteriorates in the kinetic variable, on the contrary, the regularity of the force field increases with time, since (in analytic regularity)

(3.37) F (t, 0)=0, F (t, k) = O(e−2πλ|k|t). | | b b Of course this implies the time decay of F like O(e−2πλt), but (3.37) is much more precise by keeping track of the respective size of the various modes. A simplistic way to summarize these apparently conflicting be- haviors is that there is deterioration of the regularity in v, improvement of the regularity in x. In the study of the linearized equation, we can live without knowing all this qualitative information, and it is not surprising that it has apparently never been recorded in a fully explicit way. But this will become crucial in our analysis of the nonlinear equation. 52 3. LINEARIZED VLASOV EQUATION NEAR HOMOGENEITY

3.7. Appendix: The Plemelj formula The Plemelj formula states that, in ′(R), D 1 1 (3.38) = p.v. + iπ δ ; x i0 x 0 −   or equivalently, 1 1 (3.39) = p.v. iπ δ . x + i0 x − 0   Let us recall that 1 1 p.v. = lim |x|≥ε . x ε→0 x   A more explicit expression can be given for this limit. Let χ be an even function on R with χ(0) = 1, χ Lip L1(R); in particular ∈ ∩ 1|x|≥ε χ(x) dx/x = 0 and (1 χ(x))/x is bounded. Then for any ϕ Lip(R) L1(R), − R ∈ ∩ (3.40) ϕ(x) ϕ(x) ϕ(x) dx = χ(x) dx + (1 χ(x)) dx x x x − Z|x|≥ε Z|x|≥ε Z|x|≥ε ϕ(x) ϕ(0) (1 χ(x)) = − χ(x) dx + ϕ(x) − dx x x Z|x|≥ε   Z|x|≥ε ϕ(x) ϕ(0) (1 χ(x)) − χ(x) dx + ϕ(x) − dx. −−→ε→0 x x Z   Z Then (3.38) can be rewritten in a more explicit way as follows: for any ϕ Lip(R) L1(R) and any χ satisfying the above assumptions, ∈ ∩ (3.41) ϕ(x) ϕ(x) ϕ(0) (1 χ(x)) lim dx = − χ(x) dx+ ϕ(x) − dx λ→0+ R x iλ x x Z − Z   Z + iπϕ(0). Now, let us prove the Plemelj formula. Pick up χ Lip L1(R) and write, for a given λ> 0, ∈ ∩ (3.42) ϕ(x) ϕ(x) ϕ(0) (1 χ(x)) dx = − χ(x) dx + ϕ(x) − dx x iλ x iλ x iλ Z − Z  −  Z − χ(x) + ϕ(0) dx . x iλ Z −  3.8. APPENDIX: ANALYTICITY AND REGULARITY 53

As λ 0, the first two integrals on the right-hand side of (3.42) converge→ to the first two integrals on the right-hand side of (3.40); so it only remains to check that χ(x) (3.43) dx iπ. x iλ −−−→λ→0+ Z − If (3.43) holds for some particular χ satisfying the requested condi- tions, then (3.38) follows and it implies that (3.43) holds for any such χ. So let us pick one particular χ, say χ(x)= e−x2 , which can be extended throughout the complex plane into a holomorphic function. Then since the complex integral is invariant under contour deformation, for ε> 0 we have e−x2 e−z2 dx = dz, R x iλ z iλ Z − ZCε − where Cε is the complex contour made of the straight line ( , ε), iθ −∞ − followed by the half-circle Dε = εe 0≤θ≤π, followed by the straight line (ε, ) (see Figure 3.6). {− } Next,∞ e−z2 e−z2 dz dz. z iλ −−→λ→0 z ZCε − ZCε In the latter integral, the contributions of both straight lines ( , ε) −∞ −2 and (ε, ) cancel each other by symmetry, so it only remains e−z dz/z = Dε iπ. This∞ concludes the proof. R −ε ε +0

Dε

Figure 3.6. The contour Cε, comprising two straight- lines and a half-circle Dε.

3.8. Appendix: Analyticity and regularity Analyticity is defined by the property of local summability of the Taylor series. Of course this obviously implies smoothness at all orders, but it does so in a uniform way, which can be quantified. There are two main ways to express this uniform regularity: in an informal writing, these are (i) f (n) = O(An n!); (ii) f decays exponentially fast at high frequencies. The proposition below provides a precise statement in the setting of periodic functions (the discussionb in the whole of Rd requires more care about the decay at infinity). I shall write N = 0, 1, 2,... . 0 { } 54 3. LINEARIZED VLASOV EQUATION NEAR HOMOGENEITY

Proposition 3.16. Let f : Td R be a smooth function. Then the following three properties are equivalent:→ (a) f is analytic; n! (b) λ,C > 0; x Td, n Nd, f (n)(x) C ; ∃ ∀ ∈ ∀ ∈ 0 | | ≤ λn (c) λ,C > 0; k Zd, f(k) C e−2πλ|k|. ∃ ∀ ∈ | | ≤ Proof of Proposition 3.16. I shall treat the case d = 1 and b leave the general case as an exercise. (n) n (a) (b): The convergence of the series n f (x0)(x x0) /n! ⇒ (n) n − in the ball B(x0,r0) implies f (x0) /n! C/r for any r 0 such that f (n)(x) C n!/rn. | | ≤ (b) (a) is obvious. ⇒ (b) (c): If ℓ N , then by repeated integration by parts ⇒ ∈ 0 e−2iπk·y f(k)= f(y) e−2iπk·y dy = f (ℓ)(y) dy, (2iπk)ℓ Z Z so b ℓ! f(k) C . | | ≤ (2πλ k )ℓ | | Let us choose ℓ to be theb integer closest to 2πλ k ; for ℓ large, ℓ (1 + ε)2πλ k , then | | ≤ | | ℓ! f(k) C (1 + ε)ℓ C √ℓ (1 + ε)ℓ e−ℓ C e−2πλ(1−δ)|k|, | | ≤ ℓℓ ≤ ≤ where δbis arbitrarily small and C may change from line to line. (c) (b): Pick up λ <λ, then from ⇒ 0 f (n)(x)= f(k) e2iπk·x (2iπk)n Xk b BIBLIOGRAPHICAL NOTES 55 and (c) one finds f (n)(x) C e−2πλ|k| (2π k )n | | ≤ | | k X n n! −2πλ|k| (2πλ0 k ) C n e | | ≤ λ0 n! Xk n! −2πλ|k| 2πλ0|k| C n e e ≤ λ0 Xk n! −2π(λ−λ0)|k| C n e ≤ λ0 Xk C n! . ≤ 2π(λ λ ) λn − 0 0 

Proposition 3.16 suggests two simple families of norms for analytic functions Td R, depending on a parameter λ: → n (n) λ f ∞ f λ = sup k k k k n≥0 n! and 2πλ|k| f λ = sup f(k) e . k k k | |   In both cases, λ can be interpretedb as a radius of convergence; these norms are equivalent up to an arbitrarily small loss in λ. Moreover, these formulas can be adapted to various contexts. The discussion can also be extended to Gevrey regularity, depend- ing on a parameter ν 1; Gevrey regularity can be defined either by ≥ 1/ν the property f (n) = O(Cn n!ν ), or by the property f(k) = O(e−c|k| ), and these definitions are equivalent up to constants.| | b Bibliographical notes Landau [59] solved the linearized Landau equation by using the separation of modes and the Fourier–Laplace transform. His treatment, based on the inversion of the Laplace transform, has been reproduced in countably many sources [1, 8, 15, 30, 45, 57, 65, 73, 93]. The rigorous justification is somewhat tricky because inverting a Laplace transform is not such a simple matter and involves integration over a complex contour, which has to be chosen properly; in fact Vlasov had it wrong in this respect, and Landau was the first to identify this subtlety. The first completely rigorous treatment is due to Backus [8], 56 3. LINEARIZED VLASOV EQUATION NEAR HOMOGENEITY who checked carefully the invertibility of the Laplace transform of the solution. Around the same time Penrose [86] proposed an alternative reasoning based on the general theory of the Laplace transform, and more importantly turned the intricate criterion (3.24) into the more effective condition from Section 3.4. The presentation adopted in Lemma 3.5 is centered on Fourier transform rather than Laplace transform; both are of course in the end equivalent, but focusing on Fourier transform has the advantage to only use the simpler Fourier inversion rather than the subtler Laplace inversion. Lemma 3.5 does not try to reconstruct the whole solution, but only to estimate it; this will be sufficient for the study of Lan- dau damping and decay rates. The proof shows that the quantitative estimate (3.16) is very elementary (something which is useful for the extension to the nonlinear situation), and that in the end all the tricki- ness relies in showing that the transform is well defined. The systematic use of Fourier transform was introduced in [78], following a suggestion by Sigal. (Note: in [78] the condition (ii) was mistakenly stated for 0 eξ Λ rather than < eξ Λ, however with no conse- quences≤ R on≤ the results since−∞ the correctR ≤ version of Penrose’s criterion was used there). An alternative solution consists in expressing the solution as a com- bination of generalized eigenfunctions, called Van Kampen modes [23, 65, 103]. This reduces the stability analysis to the study of a dispersion equation, but this is even more complicated to justify. Morrison [75] formalized the complete integrability property of the system, thanks to the so-called R-transform, which is related to the Laplace inversion. Counterexamples by Glassey and Scheffer, showing that there is no Landau damping in the whole space for the linearized Vlasov–Poisson equation, can be found in [38, 39]. Estimates of the decay rate at small wavelengths (large length scales) are performed in [59] or [65, Section 32]. In 1960 Penrose [86] suggested that the violation of the criterion (3.33) would lead to instability. In particular, he argued that if the distribution function has a secondary bump (is nonmonotone) then the distribution is linearly unstable at large enough scales. Conversely, the Penrose criterion implies stability under small-scale perturbations. Lin and Zeng [63] have shown that instability can arise when the Penrose criterion is violated. (Note however that for a fixed box size, (3.32) may hold true even if (3.33) is violated.) Example 3.13 is taken from [65, Problem, Section 30] (in dimension d = 3). BIBLIOGRAPHICAL NOTES 57

The interpretation of the Jeans instability can be found in [15]; it does not work quantitatively so well to predict the typical galaxy diam- eter, because galaxies are not really a continuum of stars. In a phase diagram for galaxies, the Jeans length is a “spinodal” (metastability) point, which only gives an upper bound for the phase transition regime [98]. The scattering approach to Landau damping was considered by Caglioti and Maffei [22], and amplified in my work with Mouhot [78]. A self-contained treatment of the linearized Vlasov–Poisson equation can be found in Section 3 of the latter work. Wave transforms and scattering transforms are defined and studied in [31]. Ryutov [92] mentions that interpretations of the Landau damping phenomenon were still regularly appearing fifty years after the discov- ery of this effect. The wave-particle interpretation is surveyed, some- times critically, in the works of Elkens and Escande [34, 35, 36] Belmont [12] noticed that the damping rate in the linearized equa- tion depends not only on the Landau stability condition, but also on the regularity of f 0, so that “for special distribution functions” the Landau damping rate does not govern the damping of the force.

CHAPTER 4

Nonlinear Landau damping

The damping phenomenon discovered by Landau, and considered in the previous chapter, is based on the study of the linearized Vlasov equation. But the physical model, of course, is the nonlinear equation, so the question naturally arises whether damping still holds for that model, at least in the perturbative regime, that is, near a spatially homogeneous equilibrium.

4.1. Basic concerns

0 If fi = f + εh, so that the size of the initial perturbation is ε h , then the nonlinear equation is k k ∂h (4.1) + v h + F [h] f 0 + ε F [h] h =0, ∂t · ∇x · ∇v · ∇v while the linearized equation is ∂h (4.2) + v h + F [h] f 0 =0. ∂t · ∇x · ∇v Thus the long-time analysis of the linearized equation consists in performing first the limit ε 0, then the limit t . But the phys- ically relevant question is the→ opposite: look at the→∞ long-time behavior t for the nonlinear equation, and then wonder about the limit ε →0. ∞ There is no a priori reason why these two limits could be ex- changed,→ and this casts doubts on the relevance of the long-time analy- sis of the linearized equation. In large time, small cumulated nonlinear effects might lead to significant departure from the linearized equation. Moreover, the physics underlying the linear and nonlinear equa- tions may be completely different. This is apparent for instance in the analysis of conservation laws: the conservation of total energy for the nonlinear equation is lost in the linearization, and only remains the conservation of kinetic energy. Even more striking, the conservation of all nonlinear integrals C(f) dxdv is replaced, in the linearized setting, by the conservation of all position-averages h(x, v) dx. The constraints being different,RR we may expect the long-time behavior to be radically different also. R 59 60 4. NONLINEAR LANDAU DAMPING

If we look at the linearization process with the eyes of an analyst, another concern arises: the term ε F [h] vh in (4.1) is of dominant order as far as velocity derivatives are concerned.· ∇ In general, cutting dominant order terms in partial differential equations may have a dev- astating effect, which is a further reason to be doubtful.

4.2. Backus’s objection In 1960 Backus performed the first truly rigorous study of the lin- earized analysis of the Landau damping, and at the same time con- sidered the relevance of the linearization. He noted that the approxi- mation leading from (4.1) to (4.2) is natural if ε vh can be neglected 0 ∇ in front of vf in some sense. While this may seem a harmful as- sumption in∇ view of the small coefficient ε, it is in fact totally unlikely to remain true after a time O(1/ε), since it is violated even for the linearized problem. Indeed, because of oscillations of the distribution function, vh will typically become unbounded (unless we use a weak norm, but∇ we are in a context where smoothness matters much). To see this gradient growth, let us consider just the simpler free transport: if ∂ h + v h = 0, then t · ∇x h(t, k, η)=2iπη h(t, k, η)=2iπη h (k, η + kt). ∇v i Then, even if εhi is of size ε 1, the choice η kt shows that g ≪e e≃ − ε sup vh(t, k, η) const.ε k t η ∇ ≥ | |

1 as t + , and a similar g bound holds for ε vh L (dx dv). Asa consequence,→ ∞ if we wait long enough (t 1/ε),k∇ therek will necessarily come a time when the linearization postulate≫ is no longer satisfied! Of course, F [h] h might still decay in time, because we expect · ∇v F [h] to decay exponentially fast, and vh to grow only linearly. But 0 ∇ then, the linear term F [h] vf should be even “more negligible”, and it is unclear why we should·∇ care less about the nonlinear term than about the linear one... although the latter played a key role in the analysis of the linearized stability.

4.3. Nonlinear time scale While Backus’s objection provides an upper bound on the time up to which our reasoning might be correct, it does not say that nonlinear effects will eventually occur in large time, just that a naive approach to linearization is not convincing. We may now ask if there is a time scale which is naturally asso- ciated with the nonlinear effects. To make a guess, let us look for a 4.4. ELUSIVE BOUNDS 61 scale invariance of the Vlasov equation, as when one tries to evaluate, by dimensional analysis, time scales associated with various kinds of equations. So let us assume that f =1+ h solves the Vlasov equation (forget the fact that f has infinite mass), and set 1+dν θ ν fε(t, x, v)=1+ ε h ε t, x, ε v , where ν and θ are unknown parameters. (We cannot
rescale in x since we work with periodic boundary conditions.) Note that fε 1 is of size 1 − ε in L norm, and (fε 1) dv = O(ε); this normalization is natural since it is through the velocity− average that the kinetic distribution acts on itself. ThenR ∂ f + v f + F [f ] f t ε · ∇x ε ε · ∇v ε = ε1+dν εθ ∂ h + ε−ν (v h)+ ε1+ν (F h) (εθ t, x, εν v), t · ∇x · ∇v h i so fε solves the Vlasov equation if θ = ν =1+ ν, i.e. θ =1/2= ν. In other words, − −

1− d v (4.3) f (t, x, v)=1+ ε 2 h √ε t, x, ε √ε   also solves the Vlasov equation. This suggests the typical nonlinear time scale O(1/√ε), where ε is the size of the perturbation. This O(1/√ε) time scale is actually well-known in physics, and often called the O’Neil time scale: after this time scale, the nonlinear effects should be taken into account. (They may not necessarily change the qualitative long-time behavior, but they cannot be neglected.) This seems to be well satisfied in numerical experiments. The problem arises not only for the study of the Landau damping, but also already for the a priori simpler stability problem.

4.4. Elusive bounds

The study of the linearized Vlasov equation ∂th + v xh + F [h] 0 · ∇ · vf = 0 showed that the expected decay rate depends (among other ∇ 0 things) on the smoothness of f . In the nonlinear case we have ∂tf + 0 v xf + F [f] vf = 0, so the uniformly smooth background f (v) is· replaced ∇ by· the ∇ time-dependent distribution f(t, x, v). The point which may cause much worry is that even if f(t, ) is analytic, there is no hope that it satisfies smoothness estimates which· are uniform in time: fast oscillations in the velocity variable, a phenomenon which is also known as filamentation in phase space, will imply the blow-up of all regularity bounds of f in large time. 62 4. NONLINEAR LANDAU DAMPING

Then how can one hope to adapt the tools on which the linearized study was based??

4.5. Numerical simulations Numerical simulations about the large-time behavior of the non- linear Vlasov–Poisson equation are nonconclusive because of the diffi- culties in getting reliable simulations on very large times. If ε is the size of the perturbation, nonlinear effects start to appear at time scale O(1/√ε), and then tiny numerical errors cumulated over very large times can be dreadful. To summarize the situation, one can say that for a slight perturbation of the equilibrium, numerical schemes do display• the Landau damping phenomenon for large times, and some of them continue to display damping at very large times, while other ones present tiny bumps of the electric field, which sometimes do not vanish as t ; →∞ for a larger perturbation of the equilibrium, numerical schemes agree• that damping may be replaced by a much more complicated behavior, leading to a persistent electric field. Some authors claim to observe BGK waves in very large times, while others remain more cautious. Francis Filbet kindly accepted to do a few accurate simulations for me in the perturbative regime, with different methods; they led to different results, but the one that was supposed to be the most ac displayed damping at spectacular precision (more than 20 orders of magnitude for the amplitude of the electric field, and at times so large that the nonlinear effects can definitely not be neglected; see Figures 4.1 and 4.2).

4.6. Theorem Some of the previous questions are solved by the following theorem by Mouhot and myself. If n is a multi-integer and f a function I shall write f (n) = nf = ∂n1 ...∂nd f. ∇ 1 d Theorem 4.1. Let f 0 = f 0(v) be an analytic profile satisfying the Penrose linear stability condition. Further assume that the interaction potential W satisfies 1 (4.4) W (k)= O . k 2 | |  Then one has nonlinear stabilityc and nonlinear damping close to f 0. 4.6. THEOREM 63

-6 -5 d = 0.001 d = 0.001 -8 d = 0.002 d = 0.002 -10 d = 0.004 -10 d = 0.004 d = 0.006 d = 0.006 -12 d = 0.008 d = 0.008 d = 0.010 -15 d = 0.010 -14 d = 0.012 -16 -20 -18

log(|| E(.) ||) -20 log(|| E(.) ||) -25 -22 -24 -30 -26 -28 -35 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 t t

Figure 4.1. Large time behavior of the logarithm of the norm of the electric field, with two different numer- ical methods — the second one is supposed to be more accurate. The interaction is gravitational, the initial da- tum is a Gaussian multiplied by 1 + ε cos(2πkx).

0.9 0.9 d = 0.001 d = 0.001 0.8 d = 0.002 0.8 d = 0.002 d = 0.004 d = 0.004 0.7 d = 0.006 0.7 d = 0.006 0.6 d = 0.008 0.6 d = 0.008 d = 0.010 d = 0.010 0.5 d = 0.012 0.5 d = 0.012 0.4 0.4 0.3 0.3 0.2 0.2

log(|| E(.) ||) - El(.) 0.1 log(|| E(.) ||) - El(.) 0.1 0 0 -0.1 -0.1 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 3 3.5 4 t d^1/2 t

Figure 4.2. With the more accurate method from Fig- ure 4.1, a plot of log( ENL / EL ), the logarithmic ratio of the norm of the nonlineark k k electrick field to the norm of the linearized electric field. On the left the time-scale is 1, on the right the time scale is 1/√ε. Here we see that we arrive in a time regime where the nonlinearity can definitely not be neglected.

More precisely, assume that (a) f 0 is analytic in a strip of width λ0 > 0, in the sense that λn 0 −2πλ0|η| 0 n 0 f (η) C e , f 1 C ; | | ≤ 0 n! k∇v kL (dv) ≤ 0 Nd nX∈ e 64 4. NONLINEAR LANDAU DAMPING

0 (b) the initial condition fi is a perturbation of f in a strip of width λ> 0, in the sense that f f (k, η) ε e−2πµ|k| e−2πλ|η|, f (x, v) f 0(v) e2πβ|v| dxdv ε | i− 0| ≤ | i − | ≤ Z fore alle k Zd, η Rd and for some λ,µ > 0, β > 0. ∈ ∈ (c) the generalized Penrose linear stability condition is satisfied in a 0 2 0 2 strip of width λL, in the sense that, if K (t, k)= 4π W (k) f (kt) k t, then − | | (4.5) 0 eξ λ k = (K0)L(ξ) 1 κ>c 0. e ≤ R ≤ L| | ⇒ − ≥ Let then f(t, ) be the unique smooth solution of the nonlinear Vlasov · equation with interaction W and initial datum fi; let us denote by ′ F = F [f] the associated force field. Then for any λ < min(λ0,λL,λ) there is ε = ε (λ ,λ ,λ,λ′,µ,κ,C , β) such that if ε ε then ∗ ∗ 0 L 0 ≤ ∗ ′ F (t, ) = O(ε e−2πλ t) as t + , k · k → ∞ and ρ(t, x) converges (strongly and exponentially fast) to the average ρ∞ = fi(x, v) dxdv. Moreover, RR t→±∞ f(t, ) f±∞, · −−−−→weakly where f±∞ = f±∞(v) is an analytic homogeneous equilibrium; and t→±∞ f(t, ) f±∞, h · i −−−−→strongly where stands for the x-average. h·i Remarks 4.2. Here is a long series of remarks: 1. The condition W (k)= O(1/ k 2) seems critical, and only appears in the study of the nonlinear problem.| | It would somehow be easier to handle a decreasec like O(1/ k 2+δ); this may be a coincidence or some deep thing. The same amount| | of singularity is critical in the classical proofs of existence for the Vlasov–Poisson equation, because they work with just L∞ estimates for the density, which implies by Poisson coupling a (critical) log Lipschitz condition on the force field. 0 2. The analytic norm used for fi f is the most naive norm controlling exponential localization in Fourier− space and in physical space. Also the smallness restriction on the size of ε is natural. So there is not much to complain about in the assumptions of the theorem, if one accepts to work in analytic regularity and periodic boundary conditions. 4.6. THEOREM 65

3. The decay of the force field is the damping phenomenon; the existence of limit distributions f+∞ and f−∞ is a bonus. These distri- butions are not determined by variational principles (at least, not that I know). When f+∞ = f−∞ (as in the case of the linearized Vlasov equation), one may call the solution homoclinic. One can show that in general f+∞ = f−∞, in which case the solution may be called hete- roclinic. 6 4. It is striking to see that a whole neighborhood of a stable equi- librium f 0 is filled by homoclinic/heteroclinic trajectories. This is not predicted by the (random) quasilinear theory of the Vlasov–Poisson equation, neither by the statistical theory: there is in fact no random- ness in Theorem 4.1. This abundance of homoclinic/heteroclinic orbits is possible only because, thanks to the infinite dimension, one can play with the various topologies. The fact that the conditions for damping are expressed in terms of the initial condition alone is a considerable improvement of previous results in the field. 5. The proof provides a constructive approach of the long-time behavior of f, which makes it possible to exchange the limits ε 0 and t , perform asymptotic expansions, etc. In particular,→ one → ∞ can check that the asymptotic state f+∞ “keeps the memory” of the initial datum, in a way that goes beyond the invariants of motion. The mere existence of heteroclinic trajectories demonstrates this, since the invariants (energy, nonlinear integrals) do not distinguish between the two directions of time. At first sight this confirms an objection raised by Isichenko in 1966 against the statistical theory of the nonlinear Vlasov equation. However, on second thoughts, the statistical theory can counterattack, because of the high regularity which is involved in the result. Theorem 4.1 is based on an analytic regularity; even if this is later relaxed in a Sobolev or even Cr regularity, these classes of regularity are probably negligible in a statistical context, where typical distributions are probably not smooth. 6. Implicit in Theorem 4.1 is a nontrivial perturbative large-time existence result, which in dimension 3 is covered neither by the Lions– Perthame theory (because this is a periodic setting) nor by the Pfaffelmoser– Batt–Rein theory (because the analyticity assumption is not compati- ble with a compact support). It does fall within the range of the Horst theory; however, that theory only applies up to dimension 3 for Pois- son interaction, while our result applies in any dimension. It should be noted that there is no contradiction with known blow-up results in di- mension 4, because the latter assume that the total energy is negative (and that the problem is set in the whole space), while our assumptions 66 4. NONLINEAR LANDAU DAMPING of weak inhomogeneity automatically imply that the energy is positive. (Nobody knows if there is a simple criterion such as positivity of energy to guarantee global existence in any dimension.) 7. An important corollary of the theorem is the orbital stability in s 2 a strong sense: if Hx,v stands for the L -Sobolev space of order s on Td Rd x v, then under the assumptions of Theorem 4.1, for any s> 0 we have× (4.6) f(t, x + vt, v) f 0(v) = O(ε), s − Hx,v uniformly as t + . Since L2(dxdv) is invariant under the action of → ∞ 0 free transport, the norms in (4.6) control f(t, ) f 2 Rd Td . Using k · − kL ( × ) moment bounds and Sobolev injections, one may also control f f 0 in Lp norms, for all p 1. − While such an L≥p stability is much weaker than Landau damping, nobody knows how to establish it without going to the full statement of Landau damping, or without using regularity bounds — except in the case of a monotone distribution for Coulomb interaction, which can be handled without regularity by much simpler Lyapunov functional techniques. 8. The critical regularity to which the proof applies is the Gevrey class ν , for any ν < 3 in the favorable case when K(t, k) dt < 1; this appliesG for instance for gravitational interaction be| low the| Jeans R length. More generally, if the interaction satisfies W (k) = O(1/ k 1+γ) and K(t, k) dt < 1 then the critical regularity exponent| | for the| proof| | | to work is ν = γ + 2. Here is a precise statement wherec we do not care aboutR the critical exponent: Theorem 4.3 (Nonlinear Landau damping in Gevrey regularity). Let W : Td R be an interaction potential such that W (k) = O(1/ k 2). Let→f 0 = f 0(v) be an analytic profile such that | | | | λn c 0 −2πλ0|η| 0 n 0 f (η) C e , f 1 C , | | ≤ 0 n! k∇v kL (dv) ≤ 0 n X for some λe0 > 0, and satisfying the Penrose stability condition with Landau width λ > 0. Then there is θ > 0 such that for any ν L ∈ (1, 1+ θ), β > 0, α< 1/ν, there is ε∗ > 0 such that if

1/ν 1/ν ε := sup (f f 0)(k, η) eλ|η| eλ|k| + f (x, v) f 0(v) eβ|v| dv dx i− | i − | k,η ZZ   satisfies ε eε∗,e then there is c> 0 such that the solution f = f(t, x, v) of the nonlinear≤ Vlasov equation with interaction W and initial datum 4.7. THE INFORMATION CASCADE 67 fi satisfies α F [f](t, )= O ε e−ct , · and f(t, ) converges weakly to some asymptotic
Gevrey profile f±∞ = f (v) as· t , with convergence rate O(ε e−ctα ). ±∞ → ±∞

4.7. The information cascade How can one reconcile the reversibility of the nonlinear Vlasov equa- tion and the convergence in large time? Convergence seems to be about loss of information, which should go with an increase of entropy; but we have seen that the Vlasov equation preserves the entropy. So what? The solution to this apparent paradox (well understood by some physicists at least fifty years ago, but still mysterious to many others, even today) is that all the information goes away to high frequencies, where it is hidden, becoming inaccessible. The following numerical simulations will illustrate this.

4e-05 4e-05 t = 0.16 t = 2.00 3e-05 3e-05

2e-05 2e-05

1e-05 1e-05

0 0 h(v) h(v)

-1e-05 -1e-05

-2e-05 -2e-05

-3e-05 -3e-05

-4e-05 -4e-05 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 v v

Figure 4.3. A plot of (f f 0)(t, x, v) for some fixed x, as a function of v, at two different− times t. The interac- tion is gravitational. Notice the fast oscillations of the distribution function, which are very difficult to capture by an experiment.

In fact, the cascade of energy, already present in the free transport evolution, still holds: energy (or information) flows from low to high frequencies. The complete integrability has been lost, but the energy transfer still holds. This is similar in spirit to the KAM phenomenon (KAM = Kolmogorov–Arnold–Moser), to which I shall come back later; for the moment let me just mention that there are common points and differences with the KAM theory. 68 4. NONLINEAR LANDAU DAMPING

-4 -12 electric energy in log scale gravitational energy in log scale -5 -13 -14 -6 -15 -7 -16 -8 -17 -9 -18 log(Et(t)) log(Et(t)) -19 -10 -20 -11 -21 -12 -22 -13 -23 0 10 20 30 40 50 60 70 80 90 100 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 t t

Figure 4.4. Time-evolution of the norm of the field, for electrostatic (on the left) and gravitational (on the right) interactions. In the electrostatic case, the fast time- oscillations are called Langmuir oscillations, and should not be mistaken with the velocity oscillations.

To go back to the information, let us note that f f log f = ρ log ρ + f log , ρ ZZ Z ZZ and the first term on the right-hand side converges to 0 because ρ strongly converges to a constant. So all the information becomes “ki- netic” in the limit. (Due to the oscillations, a priori we cannot pass to the limit; but that becomes possible if we go along trajectories of the free transport and use the gliding regularity.) A final remark is in order: because of the time-reversibility, any stability result, read backwards in time, should imply an instability result. This is true, however with a catch on the topologies involved. Landau damping asserts convergence in large time in the weak topol- ogy, when the initial datum is perturbed in the strong topology. Read- ing it backwards implies a instability in the strong topology, when the initial datum is perturbed in the weak topology. In particular, Landau damping is by no means in contradiction with the time-reversibility.

4.8. Scheme for attack Suppose we want to solve the nonlinear Vlasov equation by an iter- ative scheme. The most intuitive idea is to consider that particles are driven by the field created by the other particles, so each approxima- tion f n of the density generates a force field, which is used to drive the 4.8. SCHEME FOR ATTACK 69 next approximation f n+1: ∂f n+1 + v f n+1 + F [f n] f n+1 =0. ∂t · ∇x · ∇v While this natural quasilinear scheme makes perfect sense for the short-time analysis, it is poorly adapted to the long-time study. Indeed, recall that at the linear level, the damping comes from f being viewed as the driving force (the term F [h] in (3.4)), not as the driven distribution. So the next attempt would be to write ∂f n+1 + v f n+1 + F [f n+1] f n =0. ∂t · ∇x · ∇v n The problem with this is that a higher order term vf has been treated in a fully explicit way, as if it were a perturbation;∇ in particular, a deterioration of the short-time regularity is to be expected. A more promising approach is as follows. Let us linearize the Vlasov equation around some solution f(t, x, v), not necessarily stationary: if h is the first order variation, it satisfies ∂h + v h + F [f] h = F [h] f. ∂t · ∇x · ∇v − · ∇v Good estimates on the solution to this linearized equation would sug- gest the possibility to perturbatively estimate the nonlinear Vlasov equation. To get cleanly rid of the higher order term h, let us use ∇v the method of characteristics: if (Xτ,t(x, v),Vτ,t(x, v)) stands for the position at time t in phase space of a particle which is in state (x, v) at time τ, then d h t, X (x, v),V (x, v) = F [h] f t, X (x, v),V (x, v) . dt 0,t 0,t − · ∇ 0,t 0,t So  
 

(4.7) h(t, x, v)= h 0,Xt,0(x, v),Vt,0(x, v)  t  F [h] vf τ,Xt,τ (x, v),Vt,τ (x, v) dτ. − 0 · ∇ Z   The representation (4.7) naturally splits
the problem into two parts: estimate the characteristics; • estimate the reaction equation (4.7) along characteristics. • One main driving idea is the following: if characteristics are close to free transport trajectories, then the reaction equation will produce an asymptotically vanishing force field, which in turn implies that tra- jectories are close to free transport. The goal is to make quantitative sense of this intuition and turn it into a “positive” loop. A first step is 70 4. NONLINEAR LANDAU DAMPING to find adapted norms, a problem which will be considered in the next section.

Bibliographical notes Backus [8] was certainly the first one to point out the conceptual problem caused by the interversion of the asymptotic regimes t 0 → ∞ and fi f 0. Thus his paper contains both positive state- mentsk (the− firstk mathematically → rigorous treatment of the linear Landau damping) and expression of skepticism, which has not been answered until [78]. The nonlinear time scale O(1/√ε) was predicted in 1965 by O’Neil [85] with a more sophisticated argument than the naive scaling ap- proach presented in Section 4.3, and is anyway well checked in numer- ical simulations. Even if O(1/√ε) is the time scale where significant departure ap- pears between the solutions of the linear and nonlinear equations; it does not mean that the qualitative behavior of these solutions differ much. Actually, O’Neil argues that damping still holds true for larger time scales, even though not infinite. He also stumbles on the problem pointed out by Backus. Numerical investigation of the Landau damping, for small and larger perturbations, was performed by several authors at the end of the nineties, when precise methods started to be available [71, 110]. Since then, more efficient schemes have become available [46]. All pictures reproduced in this chapter have been prepared by Filbet. The mysterious but popular quasilinear theory is presented in a number of sources [65, Section 49] [1, Section 9.1.2] [57, Chapter 10], with more or less convincing arguments. Putting this theory on a decent level of rigor looks like a challenge. Isichenko’s criticism of the statistical theory of Vlasov equation ap- pears in [54]. Using an analogy with a random potential problem, he argues that the convergence to equilibrium should be slow; however the analogy used by Isichenko is not so sound, and in fact very fast convergence can occur, as demonstrated in [22] and in the present pa- per. Isichenko’s paper is still worth reading for his interesting insights, though. Nonlinear stability of monotone homogeneous profiles for the elec- trostatic Vlasov–Poisson equation was studied by Rein [90]. The two- stream instability has been established by Guo and Strauss for large bumps [43]. For nonmonotone distributions with small bumps, when the Penrose stability criterion is satisfied, the nonlinear stability is BIBLIOGRAPHICAL NOTES 71 much trickier and it was suggested in [48] that it could fail; Theorem 4.1 shows that this is not the case, at least in analytic regularity. Caglioti and Maffei [22] were the first to construct some exponen- tially damped solutions to the Vlasov–Poisson equation (in dimen- sion 1); they also noted that this implies, by time-reversibility, the instability in the weak topology. Another construction to damped so- lutions was performed by Hwang and Vel´azquez [53]. Theorem 4.1 is taken from [78], as well as the comments made right after its statement. The discussion in Section 4.8 follows the same source.

CHAPTER 5

Gliding analytic regularity

Analytic regularity may seem very specific, but it is for sure the first setting to understand in the study of Landau damping, for phys- ical reasons (because it is associated to exponential damping) and for historical reasons as well (because this is the case that was treated by Landau and all those following his steps). But the obstacles related to the study of the limit t require much care in the choice of functional space. → ∞

5.1. Preliminary analysis There are many families of analytic norms for kinetic distributions f(x, v). A particularly simple family is defined by the formula

2πµ|k| 2πλ|η| (5.1) f λ,µ = sup f(k, η) e e . k k k,η

As in the last Appendix of Chapter e 3 (Section 3.8) one can interpret λ and µ as a width of analyticity strip in the v and x variables, respec- tively. To evaluate the relevance of these functional spaces, let us first see how they behave under the transport equation

(5.2) ∂ f n+1 + v f n+1 + F [f n] f n+1 =0, t · ∇x · ∇v which amounts to let particles evolve at stage n + 1 in the force field created by the distribution at stage n (as in Section 4.8). By the method of characteristics,

n+1 n n f (t, x, v)= fi Xt,0(x, v),Vt,0(x, v) ,

n n   where (Xt,0(x, v),Vt,0(x, v)) is the initial state of particles evolving in the force field induced by f n, which at time t will be in state (x, v). Thus one is naturally led to study the behavior of norms with respect to composition. But these norms do behave very badly: to fix ideas, let us assume that St(x, v) satisfies “perfect” estimates, as good as (say) 73 74 5.GLIDINGANALYTICREGULARITY

2 Id ; and observe that k η f (2Id) =2−2d sup f , e2πµ|k| e2πλ|η| ◦ λ,µ 2 2 k,η  

=2−2d sup fe(k, η) e2π (2µ)|k| e2π(2λ)|η| k,η −2d =2 f 2 λ,e2µ. k k In other words, the norms λ,µ are not stable under composition by smooth diffeomorphisms, ink·k sharp contrast with Sobolev or Cr norms. Now imagine the disaster: each time one iterates the estimates, one loses a factor 2 on the width of the analyticity strip, so that there is nothing left in the end... How does our norm behave in the alternative scheme ∂ f n+1 + v f n+1 + F [f n+1] f n =0, t · ∇x · ∇v also considered in Section 4.8? First, it seems important to record the fact that this scheme loses a derivative, and the norms λ,µ do not perform this naturally. k·k Then, because of filamentation, the best one can hope for f is an estimate in the style of the solution of free transport g(k, η + kt): in the best of worlds, f(t, k, η) C e−2πλ|η+kt| e−2πµ|k|. e ≤ 2πλt But then f λ,µ ’ e (choose η = kt, k = 1); and even worse, k2πλk|k|t e − | | f λ,µ ’ e for any k such that the mode k does not vanish. If kallk modes k are represented, one expects the norm of f to grow faster than any exponential! This of course is a disaster for the large-time analysis. 5.2. Algebra norms Among all families of analytic norms, two deserve a special mention; let us present them in dimension 1: n 2πλ|k| λ (n) (5.3) f F λ = e f(k) f Cλ = f L∞ , k k Z | | k k N n! k k Xk∈ nX∈ 0 (n) b where f stands for the derivative of order n of f, and N0 = 0, 1, 2,... . The first norm (as it is written) makes sense only for periodic{functions,} while the second one makes sense for any smooth function on R. λ λ Proposition 5.1. With λ standing either for the or the norm, one has k·k F C fg f g . k kλ ≤k kλ k kλ 5.2. ALGEBRA NORMS 75

Sketch of proof. Let us prove the inequality for, say, the λ norm: the Leibniz differentiation formula implies C

n n λ (n) λ n! (m) (n−m) (fg) f ∞ g ∞ n! L∞ ≤ n! m!(n m)! k kL k kL n n m≤n X X X − m+m′ λ (m) (m′) ∞ ∞ = ′ f L g L ′ m! m ! k k k k m,mX m m′ λ (m) λ (m′) = f ∞ g ∞ . m! k kL m′! k kL m ! ′ ! X Xm 

n n As an immediate corollary, we have f λ f λ. This remarkable algebra property implies good propertiesk withk ≤k respectk to composition as well: there will be a loss of exponent (that is unavoidable), but it will be controlled.

λ λ Proposition 5.2. With λ standing either for the or the norm, one has k·k F C

f (Id + G) f ν, ν = λ + G λ. ◦ λ ≤k k k k

Note carefully: the constant in front of the right-hand side is equal to 1; and the size of G (in the λ-norm) is found in the regularity index on the right-hand side.

Proof of Proposition 5.2. Let us do it for the λ norm. Writ- ing h = f (Id + G), in the sense of formal series we haveC ◦

f (n)(x) h(x)= Gn(x), n! n X whence

m! h(m)(x)= f (n+k)(x)(Gn)(ℓ)(x), k! ℓ! n! n k+Xℓ=m X 76 5.GLIDINGANALYTICREGULARITY so m k+ℓ λ (m) λ (n+k) n (ℓ) h ∞ f ∞ (G ) ∞ m! k kL ≤ k! ℓ! n! k kL k kL m m n X X k+Xℓ=m X k+ℓ λ (n+k) n (ℓ) = f ∞ (G ) ∞ k! ℓ! n! k kL k kL k,ℓ,nX k λ n (n+k) G f ∞ ≤ k! n! k kλ k kL Xk,n (r) r! f ∞ λk G n k kL ≤ k! n! k kλ r! r k+n=r ! X X r λ + G λ (r) k k f ∞ ≤ r! k kL r
X = f , k kλ+kGkλ where Newton’s binomial formula was used.  Proposition 5.2 admits some variants: in particular, it is possible to mix norms:

(5.4) f (Id + G) f F ν , ν = λ + G C˙λ , ◦ Cλ ≤k k k k

˙ λ λ where the seminorm is obtained from the norm by throwing away the zero mode.C (The proof of (5.4) is a consequenceC of the dreaded Fa`a di Bruno formula and will be a pleasant exercise for the reader.) Working in kinetic theory, it is particularly convenient to hybridize the two spaces: apply the recipe to the velocity space and the recipe to the position space. (Note inC particular that for the regularity of characteristicsF there is hardly any choice: since these are unbounded functions of v, it would anyway be very tricky to apply them a Fourier- based method.) With this “hybrid” choice, the recipe will take advantage of the periodic geometry of the position space, andF property (5.4) will guar- antee that composition by the characteristics is still properly handled. Let us also generalize to d dimensions, add a parameter γ to count derivatives (for technical reasons, this is needed only in the x variable), and another parameter p to modulate the integrability; we are led to the formula n 2πµ|k| γ λ n (5.5) f λ,(µ,γ);p = e (1 + k ) f(k, v) , k kZ | | n! ∇v Lp(dv) k∈Zd n∈Nd X X0 b 5.3. GLIDING REGULARITY 77 where f(k, v)= f(x, v) e−2iπk·x dx is the Fourier transform of f in the x variable only. Then we have good properties generalizing Property R 5.1 suchb as 1 1 1 fg λ,(µ,γ);p f λ,(µ,γ);q g λ,(µ,γ);r , = + . k kZ ≤k kZ k kZ r p q

5.3. Gliding regularity Now it remains to take into account filamentation, that is, the ap- pearance of fast oscillations in the velocity variable. Since we cannot avoid it, let us accept it and incorporate it in the norm. This amounts to introducing a parameter τ (time-like) and to define f = f S0 , k kZτ k ◦ −τ kZ 0 where the regularity indices are implicit, and St (x, v)=(x + vt, v) is 0 the flow associated with free transport. (So fi S−τ is the backward solution of free transport.) ◦ This provides a family of functional spaces depending on a param- eter τ, which can a priori be chosen as one wishes, the idea being that τ is equal to, or at least asymptotic to, the time of the equation. Thus we adapt our regularity scale to the filamentation; or, we focus on the relevant Fourier modes as time goes by. All in all, we are led to the final definition of the norms: for a function f = f(x, v), Z

(5.6) f λ,(µ,γ);p k kZτ n 2πµ|k| γ λ n = e (1 + k ) v +2iπτk f(k, v) . | | n! ∇ Lp(dv) k∈Zd n∈Nd 0 X X
By default, τ = 0, γ = 0 and p = . b ∞ Remarks 5.3. Here are some important remarks about the norms. Z 1. If f = f(t, x, v) is a solution of the free transport equation, then f(t, ) = f(0, ) k · kZt k · kZ (the regularity indices being implicit). λ 2. If f = f(v), then the f λ,(µ,γ) norm reduces to the norm. k kZτ C ν 3. If f = f(x), then the f λ,(µ,γ) norm reduces to the norm k kZτ F with ν =(λτ +µ,γ) (that is, the λτ+µ space with γ additional deriva- tives). The crucial point is that theF regularity in x improves with t, as it should be in view of our discussion at the end of Chapter 3. 78 5.GLIDINGANALYTICREGULARITY

4. As a final remark, we shall almost never try to compare norms τ for various values of τ, because this is very costly in the velocity Zregularity: we don’t have anything better than

f λ,µ f λ,µ+λ|τ−τ′| , Z Z k k τ′ ≤k k τ and this becomes unaffordable as soon as τ τ ′ is large. | − |

5.4. Functional analysis Now one can study the main properties of the spaces, with respect to product, composition, differentiation (the analyticZ regularity implies a control of the derivative in terms of the function itself), inversion (estimated by means of a fixed point theorem), averaging... Below are some of the main results. In the following formulas, I do not mention those indices which are similar on the left and right-hand sides of the inequalities: 1 1 1 (5.7) fg ;r f ;p g ;q , = + ; k kZ ≤k kZ k kZ r p q

(5.8) f x + X(x, v), v + V (x, v) f α,β;p, λ,µ;p Zσ Zτ ≤k k
where α = λ + V λ,µ , β = µ + λ τ σ + X σV λ,µ ; k kZτ | − | k − kZτ C (5.9) f λ,µ f λ,µ ; k∇x kZ ≤ µ µ k kZ − (5.10) 1 1+ τ 1 < λ/λ 2= vf Zλ,µ C + f λ,µ ; ≤ ⇒ k∇ k τ ≤ λ λ µ µ k kZτ  − − 

(5.11) α = α(d) > 0; (F Id ) λ′,µ′ α ∃ k∇ − kZτ ≤ −1 = F G Id λ,µ 2 F G λ,µ , ⇒ ◦ − Zτ ≤ k − kZτ ′ ′ where λ = λ +2 F G λ,µ , µ = µ +2(1+ τ ) F G λ,µ ; k − kZτ | | k − kZτ

(5.12) f dv f λ,µ;1 ; Zτ λ|τ|+µ ≤k k Z F

λ (5.13) f dx f Zτ . λ ≤k k Z C

BIBLIOGRAPHICAL NOTES 79

A more subtle inequality, which allows to “cheat” with the time parameter, is

(5.14) f x v(t τ), v dv f Zλ(1+b),µ;1 , − − λt+µ ≤k k τ− bt Z F 1+b
where b> 1 and t> 0 are given parameters.

All these− properties will be convenient to study the nonlinear Vlasov equation. One may complain about the complicated nature of the norms; but it is possible to inject these norms into more standard norms, up to an arbitrarily small loss on the regularity indices. Thus, even if we work out the estimates in the complicated norms, we will be able to state the result in the simple-minded normsZ defined by Y 2πλ|η+kt| 2πµ|k| f λ,µ := sup f(k, η) e e . Yτ k k k,η

To estimate the norms bye the norms, we have the simple inequality Y Z

(5.15) f λ,µ f λ,µ;1 . k kYτ ≤k kZτ Conversely, to estimate the norms by the norms, we have the more subtle inequalities Z Y C(d, µ) (5.16) f λ,µ;∞ f λ,µ k kZτ ≤ (λ λ)d (µ µ)d k kY τ − − and (5.17) 1 min(λ−λ,µ−µ) 2πβ|v| f λ,µ;1 C f λ,µ + f(x, v) e dv dx . k kZτ ≤ k kY τ | |  Z  The latter inequality holds as soon as λ < λ Λ, µ < µ < M, 0 < b β B, f e2πβ|v| dv dx E, and the≤ constant C only depends≤ on Λ,≤M, B, |β,| E. The mechanism≤ of proof is similar to the Sobolev injectionsR. All inequalities in this section seem rather sharp, except for (5.17), for which one may conjecture that the best constants in the right-hand side of (5.17) are polynomial (instead of exponential) in 1/ min(λ λ, µ µ). − − Bibliographical notes Algebra norms similar to those in Section 5.2 are well-known in certain circles, and appear for instance in [3]. The idea to combine them, the resulting composition formulae, and the notion of gliding 80 5.GLIDINGANALYTICREGULARITY regularity were introduced in [78]. Detailed proofs can be found in Section 4 of that work. Gliding regularity is somehow reminiscent of the philosophy used by Bourgain [19] in the definition of his Xs,b spaces, which are defined by comparison with some unperturbed reversible dynamics; a difference is the role of the time variable, which in our treatment is just a parameter. The conjecture according to which the constants in (5.17) might be polynomial is briefly discussed in [78], and an application to the study of the nonlinear stability in “low” regularity is sketched. The picture is far from clear. A baby conjecture retaining the same features would be as follows: Show that there are s,C > 0 such that for any integers r k and any function f : R R, ≤ → −s 1−ε ε f r Cε f f k , ε = r/k. k kC ≤ k kC0 k kC Mironescu gave me a proof of the related interpolation inequality 1 k+1 m k+2 k+2 f k Ck f f , k kC ≤ k kL1 k kCk+1 where any m> 1 will work; I think this provides hope for the general case. CHAPTER 6

Characteristics in damped forcing

Before turning to the nonlinear problem where the distribution function determines the force, let us address the linear problem in which the force field is given and drives the distribution function, and let us assume on the force field the desired qualitative features. Of course, the study of the transport equation can be reduced to the understand- ing of particle trajectories (characteristics), so we shall focus on those trajectories.

6.1. Damped forcing Let be given a small gradient force field F (t, ) whose analytic reg- ularity improves linearly in time: with the notation· (5.3), F (t, ) = O(ε). · F λt The question is about the qualitative behavior of trajectories; in particular, are they transient like free transport trajectories, or can they be trapped and go along complicated trajectories? To get a first feeling, let us proceed to a formal asymptotic analysis. As before, we write Xs,t(x, v), Vs,t(x, v) for respectively the position and velocity at time t, starting from time s at (x, v). Writing formally (1) 2 (2) V0,t(x, v)= v + ε v (t, x, v)+ ε v (t, x, v)+ ..., t t (1) 2 (2) X0,t(x, v)= x + vt + ε v (s, x, v) ds + ε v (s, x, v) ds + ..., 0 0 Z Z 2 we expect F (t, X0,t(x, v)) = F (t, x + vt)+ O(ε ), then the equation X¨ = F (t, X) leads to t 2 V0,t(x, v)= v + F (s, x + vs) ds + O(ε ), Z0 so we expect V0,t(x, v)= v + O(ε). If this is in an analytic norm taking derivatives into account, we expect in particular V (x, v) I = O(ε), |∇v 0,t − | so the flow should be invertible if ε is small enough. 81 82 6. CHARACTERISTICS IN DAMPED FORCING

To summarize: our guess is that the trajectories remain perturba- tions of the free flow (x + vt, v), uniformly in time.

6.2. Deflection To compare the perturbed dynamics to the free dynamics, let us 0 write St,τ =(Xt,τ ,Vt,τ ), St,τ =(x (t τ)v, v), and define the deflec- tion operator: for 0 τ t, − − ≤ ≤ (6.1) Ω = S S0 . t,τ t,τ ◦ τ,t That is, start from time τ, evolve by the free dynamics up to time t, and then evolve backwards by the perturbed dynamics, back to time τ. As t , Ωt,τ converges to what is sometimes called a wave transform; when→ ∞ one considers both asymptotics t + and t this gives rise to scattering operators. → ∞ → −∞ Proposition 6.1. With the above notation, if λ′ <λ, µ′ <µ and (µ µ′)(λ λ′)2 (6.2) F := sup F (t, ) F λt+µ − − , ||| ||| t≥0 k · k ≤ C where C is large enough, then

−2π(λ−λ′)τ 1 Ωt,τ Id λ′,µ′ C F e min t τ, . − Zτ ≤ ||| ||| − λ λ′   − Proposition 6.1 provides an analytic deflection, modulo a small loss on the regularity index; it can be generalized (e.g. by changing τ on the left within some constraints), but for the moment this is quite sufficient to give a first idea. This estimate is (a) uniform as t ; (b) small as τ →∞t; (c) exponentially→ small as τ . →∞ Assuming an initial departure O(ε) from equilibrium, so that F is of order ε. Condition (6.2) can be fulfilled only if max(λ λ′,µ µ′) is at least of order ε1/3. We shall discuss later (in the course− of Chapter− 8) how to replace this estimate by a time-dependent loss of the regularity parameter λ, with the loss vanishing in large time.

Sketch of proof of Proposition 6.1. The principle of the proof is a standard fixed point reasoning. First let us make the ansatz

S (x, v)= x v(t τ)+ Z (x, v), v + ∂ Z (x, v) t,τ − − t,τ τ t,τ   6.2. DEFLECTION 83

(the second component of St,τ is the τ-derivative of the first one). The equation on Z is ∂2Z = F τ, x v(t τ)+ Zt,τ (6.3) ∂τ 2 − −    Z =0, ∂ Z =0.  t,t τ |τ=t t,τ So Z appears as a fixed point of Ψ : W Z, where Z is the solution of 7−→ ∂2Z = F τ, x v(t τ)+ Wt,τ (6.4) ∂τ 2 − −    Z =0, ∂ Z =0.  t,t τ |τ=t t,τ For given t, Zt,τ is a function of τ [0, t]; the norm introduced in [78] is a slight variation of the following:∈

Zt,τ λ′,µ′ Zτ (Zt,τ )0≤τ≤t := sup k k [0,t] 0≤τ≤t ( R(τ, t) )

where 2 ′ (t τ) 1 R(τ, t)= C e−2π(λ−λ )τ min − , . 2 (λ λ′)2  −  Then the goal is to check that (a) Ψ(0) 1 and (b) Ψ is 1- k k[0,t] ≤ Lipschitz in the norm [0,t], on the ball of radius 2 (for the same norm). If that is true, itk· follows k by a classical fixed-point theorem that Ψ has a unique fixed point in the ball of radius 2, and this provides the desired estimate. Let us give a hint of how to perform these estimates. For (a), we see that Ψ(0) = Z0 such that t Z0 = (s τ) F s, x v(t s) ds. t,τ − − − Zτ λ′,µ′
We are estimating this in τ norm, so (recalling Remark 5.3(3)) this is trivially bounded aboveZ by t

(s τ) F (s, ) λ′τ+µ′ ds. − k · kF Zτ Since F is a gradient, for s τ we have the estimate ≥ 2π(λ′τ−λs) −2π(λ−λ′)s F (s, ) λ′τ+µ′ e F (s, ) λs+µ e F ; k · kF ≤ k · kF ≤ ||| ||| in other words, thanks to the gradient structure of F , the gliding regu- larity has been converted into a time decay. Now we obtain the bound 84 6. CHARACTERISTICS IN DAMPED FORCING on Z0 by time-integration: t ′ ′ 1 (s τ) e−2π(λ−λ )s ds C e−2π(λ−λ )τ min (t τ)2, , − ≤ − (λ λ′)2 Zτ  −  and the desired property follows easily. To check the Lipschitz property (b) is hardly more tricky: if Z = Ψ(W ), Z = Ψ(W ) then we have t Z Z = (s τ) F s, x v(t s)+W F s, x v(t s)+W ds. t,τ − t,τe f− − − − − − Zτ h

i To estimatee this we write f F s, x v(t s)+ W F s, x v(t s)+ W − − − − − 1

= xF s, x v(t s)+(1 θ)W +fθW dθ (W W ), 0 ∇ − − − · − Z   and then use the functional analysis of the spacesf (with respectf to product, composition, differentiation, and evolutionZ by free transport) to bound this. A source of loss of regularity is the composition by something which has size 1 + O( W ). Since W = O(ε/(λ λ′)2), we can absorb this loss of regularityk (duek to composition)k k into− the loss of regularity in x, if ε/(λ λ′)2 is significantly smaller than µ µ′, and this explains where condition− (6.2) comes from. −  Bibliographical notes This chapter is entirely based on [78, Section 5], in which the reader can find precise definitions and estimates. CHAPTER 7

Reaction against an oscillating background

In the past chapter we were considering the time-evolution of an unknown distribution evolving in a given force field, now we shall con- sider the dual point of view: the force will be the unknown, and the forced distribution will be given. So the equation will be ∂f (7.1) + v f + F [f](t, x) f(t, x, v)=0. ∂t · ∇x · ∇v Formally, this equation describes the evolution of a gas of particles which tries to force the distribution f, however there is a flux (or trans- mutation) of particles from distribution f to distribution f, compen- sating exactly the effect of the force. Accordingly, I will informally call (7.1) the reaction equation. We shall assume on f the same estimates as on a typical solution of a transport equation, so in large time vf will oscillate fast in the velocity variable. — as in Chapter 3. ∇

7.1. Regularity extortion For mnemonic purpose, one may interpret (7.1) saying that one is pushing against an oscillating wall, which at times takes energy and at times gives it back, so that it is not clear whether at the end of the day one gets exhausted or not. The goal of this chapter is to show that if f is quite smooth (in gliding regularity), then the force associated with f will gain regularity in time, eventually causing the exhaustion.

Proposition 7.1. Let f 0 such that f 0(η) = O(e−2πλ0|η|) and f 0 satisfies the generalized Penrose stability| condition| with stability width

λL as in Section 3.24. Assume that the interactione satisfies W (k) = O(1/ k 1+γ), γ 1. Let f = f (x, v) such that | | ≥ i i c f λ,µ;1 ε k ikZ ≤ and f(t, x, v)= f 0(v)+ h(t, x, v) with

h(t, x, v) λ,µ;1 δ, k kZt ≤ where µ > 0 and 0 <λ< min(λ0,λL). Then there are C > 0 and r,s > 0 such that the solution of (7.1) satisfies, for all λ′,µ′ with 85 86 7. REACTION AGAINST AN OSCILLATING BACKGROUND

µ′/µ < 1 and 1/2 <λ′/λ < 1,

1 δ e (λ−λ′)r F [f](t, ) F λ′t+µ′ C 1+ ′ s ε · ≤ (µ µ ) ! − for all times t 0. ≥ Remark 7.2. The assumption on the interaction potential is sat- isfied for the Coulomb or Newton interaction with γ = 1. The formu- lation above allows to discuss the influence of γ on the estimates. The rest of this chapter is devoted to a presentation of the main ingredients behind Proposition 7.1.

7.2. Solving the reaction equation Exactly as in Chapter 3, let us apply the Duhamel principle, the Fourier transform, and integrate over v: with ρ = f dv, we get R (7.2) ρ(t, k)= fi(k,kt) t 0 −2iπk·x + ( We ρ) τ, x v(t τ) ( vf )(v) e dxdvdτ b ∇ ∗ − − · ∇ Z0 ZZ t
+ ( W ρ) τ, x v(t τ) ( h) τ, x v(t τ), v e−2iπk·x dxdvdτ. ∇ ∗ − − · ∇v − − Z0 ZZ The first and second terms on
the right-hand side are
the same as in the linearized study, and the novelty is in the last integral. Since both ρ and h depend on x, and the product becomes a convolution in Fourier space, this last integral can be rewritten as (7.3) t −2iπk·x −2iπk·v(t−τ) ( W ρ) vh(τ, x, v) e e dxdvdτ 0 ∇ ∗ · ∇ Z ZZ t b −2iπk·v(t−τ) = ( W ρ) vh (τ,k,v) e dv dτ 0 ∇ ∗ · ∇ Z t Z h i = ( W ρ)b(τ,k ℓ) ( h)b(τ,ℓ,v) e−2iπv·k(t−τ) dv dτ ∇ ∗ − · ∇v 0 Zd Z Z ℓX∈ t e = W (k ℓ) ρ(τ,k ℓ) ( vh) τ,ℓ,k(t τ) dτ. 0 ∇ − − · ∇ − Z ℓ∈Zd X
The differenced with theb linearized situation is that now there are all the nonzero values of ℓ, so that the various Fourier modes of ρ are coupled to each other. To estimate the expression above, let us use the 7.2. SOLVING THE REACTION EQUATION 87 bound W (k ℓ)= O(1/ k ℓ γ) and the gliding regularity assumption ∇ − | − | on h: using spaces for simplicity, we shall assume Y d h(s,k,η) C δ e−2πλ|η+ks| e−2πµ|k|. | | ≤ 2π(λt+µ)|k| Plug this into (7.3),e take norms, multiply by e and sum over k to obtain an estimate of the λt+µ norm of the last term in (7.2) by F (7.4) t 1 Cδ ρ(τ,k ℓ) e2π(λt+µ)|k| k (t τ) e−2πλ|k(t−τ)+ℓτ| e−2πµ|ℓ| dτ. k ℓ γ − | | − Z0 k6=ℓ X | − |

We note that if λ< λb, µ< µ, then

e2π(λt+µ)|k| e−2πλ|k(t−τ)+ℓτ| e−2πµ|ℓ| e−2π(µ−µ)|ℓ| e2π(λτ+µ)|k−ℓ| e−2π(λ−λ)|k(t−τ)+ℓτ|. ≤ So (7.4) is bounded by t 1 2π(λτ+µ)|k−ℓ| −2π(µ−µ)|ℓ| −2π(λ−λ)|k(t−τ)+ℓτ| Cδ γ ρ(τ,k ℓ) e e e k (t τ) dτ 0 k ℓ | − | | | − Z Xk6=ℓ | − | t b 2π(λτ+µ)|k−ℓ| 2π −(µ−µ) |k| Cδ K(t, τ) ρ(τ,k ℓ) e e 2 dτ ≤ − Z0 k,ℓ X Cδ t b K(t, τ) ρ(τ) λτ+µ dτ, ≤ (µ µ)d k kF − Z0 where the kernel K(t, τ) is defined by

−2π(λ−λ)|k(t−τ)+ℓτ| −2π( µ−µ )|ℓ| k (t τ) e e 2 (7.5) K(t, τ) = sup | | − . 1+ k ℓ γ k,ℓ | − | ! Notice, we took advantage of the fact that λ>λ to get some decay in the exponentials appearing in (7.5). Plugging these bounds into (7.2), we conclude that

t (7.6) e2π(λt+µ)|k|t ρ(t, k) K0(t τ,k) ρ(τ,k) dτ − − k Z0 X t b b A(t)+ δ ρ(τ) λτ+µ K(t, τ) dτ, ≤ k kF Z0 2πλ|k|t where A(t) = e fi(k,kt) is the contribution of the initial da- tum. | | P e 88 7. REACTION AGAINST AN OSCILLATING BACKGROUND

To appreciate the effect of the new kernel K(t, τ) let us set f 0 =0 in (7.6); then this inequality turns into a closed inequality on ϕ(t) = ρ(t, ) λt+µ : k · kF t (7.7) ρ(t) λt+µ A(t)+ C K(t, τ) ρ(τ) λτ+µ dτ. k kF ≤ k kF Z0 Let us analyze the expression (7.5). The decay in ℓ is good, the decay in k ℓ is not so good (depending on the smoothness of the interaction),| − and| the decay in k is not good at all since k(t τ)+ ℓτ can be small even though k is very large: just choose ℓ|opposite− to k| and τ =( k / k ℓ )t. Stated otherwise, in the time-integral there is a resonance| |phenomenon| − | occurring for k(t τ)+ ℓτ =0. − 7.3. Analysis of the kernel K Let us analyze the kernel appearing in (7.5). Inside the supremum there is a good decay in k(t τ)+ ℓτ , so for practical purposes one may replace the factor k |(t −τ), appearing| in front of the exponential, by 1+ ℓ τ. (This is true| | also− for ℓ =0: if f = O(1) in gliding analytic | | regularity, then in general vf = O(t), but vf = v f = O(1), where f is the spatial average∇ of f.) Then ith∇ is noti difficult∇ h toi see that K(t, τ)h isi not better than O(τ) and that its time-integral K(t, τ) dτ is not better than O(t). And this is bad news, because it suggests the possibility of a very serious instability as t , with a growthR like, say, et2 . → ∞ But let us analyze the kernel more precisely. To get an idea of its quantitative behavior, let us set d = 1, replace the slower power decay k ℓ −γ by the faster decay e−c|k−ℓ|, keep only the dominant mode ℓ| =− |1 and the modes k 0, and replace k (t τ) by 1+ τ. The resulting− approximation is,≥ up to a multiplicative| | − constant, 1 (7.8) K(t, τ) = sup (1 + τ) e−2π(λ−λ)|k(t−τ)−τ| . (k + 1)γ k=1,2,...   Below is a numerical plot of τ K(t, τ) (where the interaction factor, for various values of t, after replacing7−→ the interaction factor (k+1)−γ by the nicer e−ck; that is, the function τ K(t, τ) is plotted for various values of t. A preliminary conclusion7−→ is that in the integral equation t (7.9) ϕ(t) A(t)+ K(t, τ) ϕ(τ) dτ, ≤ Z0 for each t, only certain specific values of τ seem to count (like t/2, etc.). This may seem crazy at first, but it is exactly the same principle which 7.3. ANALYSIS OF THE KERNEL K 89

1.4 12 120 exact t = 10 exact t = 100 exact t = 1000 approx t = 10 approx t = 100 approx t = 1000

1.2 10 100

1 8 80

0.8

6 60

0.6

4 40 0.4

2 20 0.2

0 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 7.1. The kernel K(t, τ) for t = 10, t = 100, t = 1000. The curve above is an approximate envelope which can be computed analytically, but which provides disastrous estimates; observe indeed that most of the mass of the kernel K concentrates on discrete times as t becomes large.

underlies the echo phenomenon, a beautiful experiment made in the sixties by Malmberg and co-workers. Let me describe the experiment and its interpretation. Prepare your favorite plasma in your favorite lab, and at time 0 excite it by a small impulse of frequency ℓ Z, say ℓ< 0. Wait until the electric field damps, and at time τ > 0 excite∈ the plasma again by a small impulse of frequency k ℓ, with k Z, k > 0. Then sit and measure the electric field, analyzing− the streng∈ th of the mode k. Around time (k ℓ)τ (7.10) t = − , e k a spontaneous response, the echo, will be recorded. The interpretation is the following: initially disturbed, the electric field has damped, but the information is still there, hidden in the fast oscillations of the distribution function in the velocity variable. Start from a homogeneous background f 0(v), apply a pulse (sharply localized in time) oscillating at spatial frequency k, from the Vlasov equation 0 the variation in the density is proportional to F (x) vf (v), so right after the pulse the density will be roughly − ·∇ f(0+, x, v) f 0(v) 2iπθ e2iπℓ·x ℓ f 0(v), ≃ − · ∇v where θ is a small constant depending on the intensity and duration of the pulse, and I use a complex notation for simplicity (only the real part makes sense). Then the distribution evolves by damping and oscillates more and more; since the electric field is O(θ), the main contribution 90 7. REACTION AGAINST AN OSCILLATING BACKGROUND

Figure 7.2. Representation of the plasma echo experi- ment, from the pioneering paper by Malmberg, Wharton, Gould, and O’Neil. is due to free transport and we have, at time τ, f(τ, x, v) f 0(v) 2iπθ e2iπℓ·(x−vτ) ℓ f 0(v)+ O(θ2) e2iπℓ·(x−vτ). ≃ − · ∇v (The last term is due to the contribution of the force derived from the linearized Vlasov equation.) At time τ the new pulse is applied; the change in the density is proportional to F vf, where the force now oscillates at spatial frequency k ℓ:− so right· ∇ after time τ the density will be − f(τ +, x, v) f 0(v) 2iπθ e2iπℓ·(x−vτ) ℓ f 0(v)+ O(θ2) e2iπℓ·(x−vτ) ≃ − · ∇v 2iπθ′ e2iπ(k−ℓ)·x (k ℓ) f 0(v) − − · ∇v +2iπθ θ′ e2iπℓ·(x−vτ) e2iπ(k−ℓ)·x 2iπ(k ℓ) ℓτ+ 2f 0(v) ℓ,k ℓ +O(θ′ θ2). − − · ∇v · − Then again, we let it evolve freely and at a later time t we have  f(t, x, v) f 0(v) 2iπθ e2iπℓ·(x−vt) ℓ f 0(v)+ O(θ2) e2iπℓ·(x−vt) ≃ − · ∇v 2iπθ′ e2iπ(k−ℓ)·(x−v(t−τ)) (k ℓ) f 0(v)+ O((θ′)2) e2iπ(k−ℓ)·(x−v(t−τ)) − − · ∇v +2iπθ θ′ e2iπℓ·(x−vt) e2iπ(k−ℓ)·x 2iπ(k ℓ) ℓτ + 2f 0(v) ℓ,k ℓ − − · ∇v · −  + O(θ3)+ O((θ′)3) .  7.4. ANALYSIS OF THE INTEGRAL EQUATION 91

(Note that the contribution of the initial pulse to the electric field is small, so from time τ on we can assume the field to be O(θ′). Also we assume that θ and θ′ are of the same order of magnitude.) Summarizing everything, f(t, x, v) is mainly made of the superpo- sition of three spatial frequencies: at spatial frequency ℓ, there is a coefficient O(θ) e−2iπ(ℓt)·v, which is rapidly• oscillating in v (for large times) and thus does not contribute significantly the electric field; at spatial frequency k ℓ, there is a coefficient O(θ′) e−2iπ((k−ℓ)t)·v, which• is also rapidly oscillating− in v; at spatial frequency k, there is a coefficient O(θ θ′) e−2iπ[k(t−τ)+ℓτ]·v, which• does not oscillate fast in v if k(t τ)+ ℓτ 0. Then although this term is second-order, it may have an− important≃ contribution to the electric field, more than the first-order highly oscillating terms. This is the source of the echo. The beauty of the echo experiment is that it demonstrates that in the Landau damping phenomenon, the information has not disap- peared: it is still there, but hidden in the high frequency oscillations in velocity. The interaction between the two spatial frequencies ℓ and k ℓ has produced a response which can be measured: a visible mani- festation− of what was meant to remain hidden. Back to the nonlinear damping problem, here is the picture which is starting to emerge. While in the linearized Vlasov equation, each mode k was evolving independently of the other ones, in the nonlinear Vlasov equation that is not the case: there is a coupling of all modes by the interaction. Exciting one mode at some time has a nonnegli- gible consequence on other modes at later times (by echoes), but this is controlled by the integral equation, mixing estimates on all modes together.

7.4. Analysis of the integral equation Now let us come back to the analysis of the integral equation t (7.11) ϕ(t)= A(t)+ K(t, τ) ϕ(τ) dτ Z0 appearing as a variant of (7.9). From the past section, we have a bad news, namely that the kernel grows linearly in time; and a good news, namely that it concentrates on “resonant” times τ, which are not too close to t. The moral is the same as can be derived from the echo experiment: the Vlasov equation is an oscillatory system which responds with time-delay. To illustrate 92 7. REACTION AGAINST AN OSCILLATING BACKGROUND why this is a good news, let us examine a few examples of baby integral equations. a kernel that is uniformly O(τ): • t ϕ(t) A + c τϕ(τ) dτ; ≤ Z0 then this yields ϕ(t) A ect2/2, which is a disaster. ≤ a kernel whose integral is O(t), and which is spread over times: • t ϕ(t) A + c ϕ(τ) dτ; ≤ Z0 then ϕ(t) A ect, which is better. ≤ a kernel whose integral is O(t), which is concentrated at the final time:• ϕ(t) A + ctϕ(t) : ≤ this is a complete disaster, the inequality does not even prevent ϕ from becoming infinite. a kernel which is O(τ), whose integral is O(t), and which is concentrated• near the final time: t ϕ(t) A + c t ϕ(τ) dτ; ≤ Zt−1 then this still allows for quick growth (how much?). a kernel whose integral is O(t), but whose mass concentrates far away• from t: t t (7.12) ϕ(t) A + c ϕ . ≤ 2 2   Then a power series expansion suggests cn tn ϕ(t) A , ≤ 2n(n−1)/2 n X which is basically the same as Ac′(log t)2 ; one can also guess this behavior directly from (7.12). This is very good since this growth is subexponen- tial (faster than any polynomial, slower than any fractional exponen- tial). In particular, we can write ϕ(t) C A eεt, where ε is arbitrarily ≤ ε small and Cε depends on ε. Why is this good? Recall that in our case ϕ(t) is an analytic norm whose regularity index increases exponentially fast with time, 7.5. EFFECT OF SINGULAR INTERACTIONS 93

′ say ϕ(t) = ρ(t) λt+µ . Then for the force F (t) we have, for λ < λ, k kF using as usual F (t, 0) = 0,

−2π(λ−λ′)t F (t) λ′t+µ e F (t) λt+µ bk kF ≤ k kF −2π(λ−λ′)t C e ρ(t) F λt+µ ≤ k ′ k CC A e−2π(λ−λ )t eεt, ≤ ε and by choosing ε close enough we can make sure that the decay of the force is still exponential in time.

7.5. Effect of singular interactions In the previous discussion and analysis of the kernel, I have replaced the power law decay W (k) k −(1+γ) of the interaction by the expo- nential decay e−c|k| which is≃| typical| of an analytic interaction. But of course, the most interestingc cases occur when W (k) only decays like a power law, corresponding to a singularity in physical space. The most important case of all is γ = 1 (Poisson coupling).c How does this singu- larity modify the picture which we formed for an analytic interaction? To get a feeling, and appreciate the influence of the strength of the singularity, let us consider the baby kernels e−α|kt−(k+1)τ| Kγ(t, τ)=(1+ τ) sup γ . k=1,2,... (k + 1) To appreciate the long-time behavior, let us perform a time-rescaling, setting kt(θ) = tK(t, tθ) (the t factor in front is there to keep the total mass of K invariant in the rescaling). As t , the exponen- tial e−α|k−(k+1)θ| t becomes localized in a neighborhood→ ∞ of size O(1/kt) around θ = k/(k + 1), and its mass becomes 2/(α(k + 1)). Thus

kt 2 1 k δ1− 1 , t −→ α (k + 1)γ (k + 1)2 k+1 Xk where the convergence is in the weak sense on the time-interval [0, t]. This suggests the approximation 1 (7.13) Kγ (t, τ) c t δ 1− 1 t. ≃ k1+γ ( k ) Xk≥1 Examination of (7.13) shows that the lower γ is, the more the echoes accumulate near t; then we are getting closer to the (very bad) regime of instantaneous response. For pedagogical purpose, one may keep in mind the image that if one sings in a very rough church (say), the 94 7. REACTION AGAINST AN OSCILLATING BACKGROUND abundance of small echoes may blur the sound in an uncontrollable way. To evaluate the influence of the kernel (7.13), let us search once n again for an exact power series solution ϕ(t)= an t to the integral equation 1 k P1 ϕ(t)= A + c t ϕ − t . k1+γ k Xk≥1    This yields 1 1 n a0 = A, an+1 = c 1+γ 1 an. " k − k # Xk≥1   The sum inside brackets is comparable to ∞ 1 n t−(1+γ) 1 dt = B(γ, n + 1) n−γ, − t ≃ Z1   −γ where B is Euler’s Beta function. So an+1 c n an, thus an Acn/(n!)γ , and we expect ≃ ≃ cn tn (7.14) ϕ(t) const. A . ≤ n!γ n≥1 X This is subexponential for γ > 1 (which is good), but exponential for γ = 1! Since γ = 1 is the most interesting case, it is tempting to believe that we stumbled on some deep difficulty. But this is a trap: a much more precise estimate can be obtained by separating modes and esti- mating them one by one, rather than seeking for an estimate on the whole norm. Namely, if we set ϕ (t)= e2π(λt+µ)|k| ρ(t, k) , k | | then we have a system of the form b c t kt (7.15) ϕ (t) a (t)+ ϕ . k ≤ k (k + 1)γ+1 k+1 k +1   −ak −2πλ|k|t Let us assume that ak(t)= O(e e ). First we simplify the time-dependence by letting 2πλ|k|t 2πλ|k|t Ak(t)= ak(t) e , Φk(t)= ϕk(t) e . Then (7.15) becomes c t kt (7.16) Φ (t) A (t)+ Φ . k ≤ k (k + 1)γ+1 k+1 k +1   7.6. LARGE TIME ESTIMATES VIA EXPONENTIAL MOMENTS 95

(The exponential for the last term is right because (k+1)(kt/(k+1)) = kt.) Now if we get a subexponential estimate on Φk(t), this will imply an exponential decay for ϕk(t). Once again, we look for a power series, assuming that Ak is constant −ak in time, decaying like e as k ; so we make the ansatz Φk(t) = m −ak →∞ m ak,m t with ak,0 = e . As an exercice, the reader can work out the doubly recurrent estimate on the coefficients ak,m and deduce P e−am a const. A (k e−ak) km cm , k,m ≤ (m!)γ+2 whence α 1 (7.17) Φ (t) const. A e(1−α)(ckt) , α< . k ≤ ∀ γ +2 This is subexponential even for γ = 1: in fact, we have taken advantage of the fact that echoes at different values of k are asymptotically rather well separated in time. As a conclusion, as an effect of the singularity of the interaction, we expect to lose a fractional exponential on the convergence rate: if the −2πλ|k|t mode k of the source decays like e , then ϕk, the mode k of the solution, should decay like e−2πλ|k|t e(c|k|t)α . More generally, if the mode (c|k|t)α k decays like A(kt), one expects that ϕk(t) decays like A(kt) e . Then we conclude as before by absorbing the fractional exponential in a very slow exponential, at the price of a very large constant: say tα − α ε t e exp cε 1−α e . ≤ 7.6. Large time estimates via exponential
moments So far we have mainly done heuristics and power expansions, now arises the question of rigorously estimating solutions of integral equa- tions. Let us leave apart the more tricky case when the modes are decoupled, and focus on the single case when there is just one equa- tion, like (7.7). So let ϕ(t) 0 solve ≥ t ϕ(t) A + K(t, τ) ϕ(τ) dτ, ≤ Z0 where K(t, τ) is given by (7.5). To estimate ϕ in an exponential scale, we shall consider exponential moments of the kernel. The main idea is that t −εt ετ (7.18) e Kγ (t, τ) e dτ Z0 will be smaller if K favors large values of t τ. − 96 7. REACTION AGAINST AN OSCILLATING BACKGROUND

It can be shown by elementary means that for t 1, ≥ t C (7.19) e−εt K(t, τ) eετ dτ , ≤ εr tγ−1 Z0 for some constants C > 0, r> 0. The important fact is that the bound on the right-hand side of (7.19) decays as t . Let us see how to exploit this information.→∞ Let ψ(t) = B eεt. If ψ satisfies ϕ(t) < ψ(t) for0 t T ≤ ≤  t ψ(t) A + K(t, τ) ψ(τ) dτ for t T ,  ≥ ≥ Z0 then u :=ψ ϕ is positive for t T , and satisfies the inequality t − ≥ u(t) 0 K(t, τ) u(τ) dτ for t T , so u will never vanish and always remain≥ positive — this is a maximum≥ principle argument. ForR small values of t, that is, 0 t T , a crude bound, in Gronwall style, is easy: it may give a very bad≤ ≤ constant like eT 2 or so, but that remains a finite constant, whatever the choice of T . For large values of t, that is t > T , we just have to check that t A + B K(t, τ) eετ dτ B eεt. ≤ Z0 But from (7.19), the left-hand side is bounded above by B C eεt A + , εr tγ−1 and this is obviously bounded above by B eεt as soon as B A/2 and t (2BC/εr)1/(γ−1), which in turn holds as soon as T is chosen≥ large enough.≥ The estimate can be refined in many ways. Instead of exponential moments, one can consider fractional exponential moments

α α e−ε t K(t, τ) eε τ , Z which gives much better results as far as the dependence on ε is con- cerned. Also, there is a variant which covers the case when the kernel is used as a source term in the linearized Vlasov equation with a nontrivial f 0, as in (7.6). This argument is more tricky and goes not only through a maximum principle but also through L2 estimates (as in Lemma 3.5) and an inequality of Young type. To work this out, one establishes BIBLIOGRAPHICAL NOTES 97 decay estimates not only on L1 exponential moments as (7.18), but also decay estimates on L2 moments 1/2 e−2εt K(t, τ)2 e2ετ dτ , Z  and uniform bounds on dual L1 moments, ∞ sup eετ K(t, τ) e−εt dt. τ≥0 Zτ The elementary method from Lemma 3.5 can then be adapted to this tricky situation. (This is somewhat painful, but the use of the inversion of the Laplace transform would probably have been quite more painful.) Finally, there is also a variant which allows to estimate the norms of all modes separately, and thus to treat the important case γ = 1. Bibliographical notes Basically all this chapter is taken from [78, Sections 6 and 7], where more precise computations and estimates are established. (But the ef- ficiency of fractional exponential moments is not noticed in that refer- ence.) The echo experiment appears at the end of the sixties, in [69] (pre- diction) and [70] (report of experiment); I learnt it from Kiessling. In fact at first it was spatial echoes which were observed, and it is only later that temporal echoes could be produced. Nowadays they are used as an indirect way to measure the strength of irreversible phenomena going on in a plasma (defect of echo indicates dissipation!), see [96].

CHAPTER 8

Newton’s scheme

In the past two chapters we have examined the two sides of the non- linear Vlasov equation near equilibrium, first as a transport equation in a small force field whose regularity improves with time, secondly as the reaction for a gas forcing an oscillating background which is a perturba- tion of equililbrium. In both cases we studied the corresponding linear problem and obtained estimates in gliding regularity that are uniform in time, at the price of a loss of regularity, or consequently a loss on the time decay rate. (Recall that the gliding regularity automatically implies a time decay on velocity averages.) Loss of regularity in the solution of the linearized equation is en- countered in a number of problems and sometimes informally called the Nash–Moser syndrome. It was overcome in the fifties by Kol- mogorov (in the proof of his celebrated 1954 theorem of the likely stabil- ity of trajectories of perturbed completely integrable Hamiltonian sys- tems) and by Nash (in his celebrated 1956 construction of smooth iso- metric Riemannian embeddings). In both cases a key idea was to work out a perturbative analysis based on the Newton iterative scheme and use the supernatural speed of convergence of this scheme to over- come the loss of regularity. Nash also showed how to take advantage of this fast convergence to squeeze in a regularization at each stage, giving birth to what is now called the Nash–Moser method, arguably the most powerful perturbative technique known to this date. Moser used it to prove Kolmogorov’s theorem in Cr regularity. Still the mighty Newton scheme will save us again. It does not mean that this is the only solution: History has shown (for the Kolmogorov theorem and even more for the Nash theorem) that the Newton scheme can sometimes be replaced by a classical fixed point technique, applied to a clever reformulation of the problem. It turns out that the condition of the damping problem is worse than the usual Nash–Moser syndrome, because the damping problem “loses” an infinite number of “derivatives”.

99 100 8. NEWTON’S SCHEME

8.0. The classical Newton scheme The general formulation of the Newton scheme is as follows. Let be given an equation Φ(z) = 0, where the unknown z lies in R or in some Banach space, and the equation should be solved in a neighborhood V where the differential DΦ is invertible. Start from some guess z0 and solve iteratively, approximating Φ at each step by its tangent to the previous approximation. So at step n, the equation to be solved is (8.1) Φ(z )+ DΦ(z ) (z z )=0, n n · n+1 − n or equivalently z = z [DΦ(z )]−1 Φ(z ). n+1 n − n · n If zn+1 always remains in V then this procedure defines inductively a sequence (zn)n∈N. Clearly if zn converges to z, then from (8.1) we have Φ(z) = 0. The claim is that if z0 is close enough to the desired solu- tion, and if Φ is twice continuously differentiable, then the convergence occurs extremely fast. To see this, use the Taylor expansion and (8.1) to deduce (8.2) D2Φ Φ(z ) Φ(z )+ DΦ(z ) (z z ) + k k∞ z z 2 k n+1 k ≤ n n · n+1 − n 2 k n+1 − nk D2Φ = k k∞ z z 2, 2 k n+1 − nk where is the supremum norm over the domain V . k·k∞ Plugging (8.2) in the identity Φ(zn+1)+DΦ(zn+1) (zn+2 zn+1)=0 yields DΦ(z ) (z z ) ( D2Φ /2) z · z −2, whence k n+1 · n+2 − n+1 k ≤ k k∞ k n+1 − nk (DΦ)−1 D2Φ z z k k∞ k k∞ z z 2. k n+2 − n+1k ≤ 2 k n+1 − nk   Iteration of the inequality z z C z z 2 yields k n+2 − n+1k ≤ k n+1 − nk n (8.3) z z Cn z z 2 . k n+1 − nk ≤ k 1 − 0k Now if Φ(z ) is small enough, then δ := Φ(z ) DΦ(z )−1 is k 0 k k 0 kk 0 k strictly less than min(1, 1/C), z1 z0 < δ and (8.3) implies induc- tively k − k n z z Cn δ2 . k n+1 − nk ≤ Then of course (zn) converges to z, and by telescopic summation

n n n n n n n z z Cn δ2 1+ C δ2 + C2 δ2 ·4 + C3 δ2 ·4 ·8 + . . . , k n − k ≤ h i 8.1. NEWTON SCHEME FOR THE NONLINEAR VLASOV EQUATION 101 which is bounded above by 2 Cnδ2n if δ is small enough. Up to changing C, we conclude that

n (8.4) z z Cn δ2 . k n − k ≤ That is, the Newton method converges like an iterated exponential (exponential of exponential). (One often says that the convergence is quadratic to express the fact that the number of significant digits doubles at each step, or equivalently that the error at each step is essentially squared.) All this is subject to the fact that (zn) remains inside the neighborhood V where the root z belongs; but in view of the estimate (8.4), this is clearly the case if z is close enough to z0.

8.1. Newton scheme for the nonlinear Vlasov equation

Consider an evolution partial differential equation ∂tf = Q(f), where the unknown is a solution f = (f(t))t≥0 and the initial datum fi is prescribed. To cast this equation in the setting of the Newton scheme, define ∂f Φ(f)= Q(f), f(0) f . ∂t − − i   Then the equation Φ(f n)+ DΦ(f n) (f n+1 f n) = 0 means (8.5) · − ∂ f n Q(f n) + ∂ (f n+1 f n) Q′(f n) (f n+1 f n)=0 t − t − − · − h n+1 i f (0) = fi for all n. n+1 n ′ n−1 n+1 n The first equation is ∂tf = Q(f ) Q (f ) (f f ), but this is not the most convenient form. It is− best to see ·f n as made− of a series of successive layers: f n = f 0 + h1 + . . . + hn, where the unknowns hn solve ∂ hn+1 = Q′(f n) hn+1 + Q(f n) ∂ f n t · − t = Q′(f n) hn+1 + Q(f n−1 + hn) Q(f n−1) Q′(f n−1) hn , · − − · h 1 n+1 i together with the initial conditions h (0) = hi, h (0) = 0. In the case of the nonlinear Vlasov equation, the nonlinearity is n−1 n n−1 ′ n−1 n n n quadratic, so Q(f + h ) Q(f ) Q (f )h = F [h ] vh . Then we arrive at the Newton− scheme− for the nonlinear− Vlasov· ∇ equation near a spatially homogeneous equilibrium f 0. First f 0 = f 0(v) is given, then f n = f 0 + h1 + . . . + hn, where h1 solves the 102 8. NEWTON’S SCHEME linearized Vlasov equation ∂h1 + v h1 + F [h1] f 0 =0 ∂t · ∇x · ∇v (8.6)  h1(0, )= f f 0, · i − and, for any n 1, (8.7) ≥ n+1 ∂h n+1 n n+1 n+1 n n n + v xh + F [f ] vh + F [h ] vf = F [h ] vh  ∂t · ∇ · ∇ · ∇ − · ∇ hn+1(0, )=0.  · In this way the nonlinear Vlasov equation has been reduced to an  infinite list of linear equations, each of which involves a source term which is quadratic in the solution of the previous equation. The Newton scheme destroys many of the properties and invariances of the original equation, however note that it is still in divergence form, so (8.8) n 2, t 0, hn(t, x, v) dxdv =0. ∀ ≥ ∀ ≥ ZZ For n = 1 we already know that

t 0, h1(t, x, v) dxdv = (f f 0)(x, v) dxdv. ∀ ≥ i − ZZ ZZ Now the goal is to study the various layers hn. We shall do this in two stages: short time, large time.

8.2. Short time estimates Proposition 8.1. Let f 0 = f 0(v) be a spatially homogeneous pro- 0 file satisfying f Zλ0;1 C0 for some λ0 > 0, and let W be an inter- action potentialk withk W≤ L1(Td). If λ′ <λ<λ and 1 0 and T∗ > 0 such that if fi satisfies 0 f f λ,µ ε, k i − kZ ≤ then for ε ε one has ≤ ∗ n an n N, t [0, T ], h (t) λ,µ;1 Cε . ∀ ∈ ∀ ∈ ∗ k kZ ≤ Remark 8.2. In this short-time estimate, it does not matter whether we use gliding regularity or not. The Vlasov equation is a first-order nonlinear partial differential equation; for such equations there is a general method to establish short-time analytic estimates, known as the Cauchy–Kowalevskaya 8.2. SHORT TIME ESTIMATES 103 theory. For our current purposes, we do not need to explicitly appeal to that theory, and can give a self-contained treatment using the hybrid analytic norms, with a regularity index which deteriorates in time. In + the next lemma, d u/dt = lim sups↓0[u(t + s) u(t)]/s stands for the upper right-hand derivative of u. − Lemma 8.3. For t small enough, d+ f λ(t),µ(t);1 cK f λ(t),µ(t);1 , λ(t)= λ Kt, µ(t)= µ Kt. dt k kZ ≤ − k∇ kZ − − The proof of this lemma is easy and relies mainly on the identity (d/dt)e2πλ|k|t =2πλ k e2πλ|k|t. | | Sketch of proof of Proposition 8.1. Let us estimate hn(t, ) · in a norm λn−Kt,µn−Kt. We get rid of the linear transport term by using the freeZ transport semigroup, and we are left with the contribu- tion of the force term. So the estimate of the variation of the norm will involve several terms, one of which is nonlinear and involves derivatives of f n and hn+1, and one of which comes from the time-variation of the regularity index; that one is linear and proportional to hn+1 . So −k∇ k d+ (8.9) hn+1 K hn+1 + C (f n f 0) hn+1 dt k k ≤ − k∇ k k∇ − k k∇ k + C hn 2 + . . . k∇ k and all the norms are λn+1−Kt, µn+1−Kt;1. The amount of regularity lost Z with time is the same for all indices n, and if λn, µn remain bounded from below and t is small enough, then these norms control a fixed norm λ,µ;1. LetZ us see how to estimate (8.9). We shall use the shorthand

h = h λn−Kt,µn−Kt;1 and assume that k kλn k kZ h δ , k kλn ≤ n then the goal is to get recursive estimates on δn. Using (5.9)–(5.10), we relate the norm of the gradient to the norm of the function, stratifying at the same time the estimate by bounding each layer hn in a different norm (the regularity deteriorates with n, so we have more information for lower indices n). Thus we write (f n f 0) h1 + h2 + . . . k∇ − kλn+1 ≤ k∇ kλn+1 k kλn+1 h1 h2 k kλ1 + k∇ kλ2 + . . . ≤ λ λ λ λ 1 − n+1 2 − n+1 δ δ δ 1 + 2 + . . . + n . ≤ λ λ λ λ λ λ 1 − n+1 2 − n+1 n − n+1 104 8. NEWTON’S SCHEME

Assuming that the sequence (λn λn+1) is decreasing, this is grossly bounded above by − δ (8.10) n , λ λ n − n+1 and if that sum is small enoughX then the second term on the right- hand side of (8.9) (C (f n f 0) hn+1 ) is absorbed by the first one ( K hn+1 ). Afterk∇ bounding− k k∇ in a similark way the last term on the right-hand− k∇ sidek of (8.9), we end up with something like C δ2 (8.11) δ n . n+1 ≤ λ λ n − n+1 2 At this stage we can choose, say, λn λn+1 =Λ/n , where Λ is very small: that is, we allow the regularity to− decrease at each step in a con- 2 trolled way. If n δn is small enough, then the sum in (8.10) is small ′ 2 2 too, so (8.11) holds true, and the recursion relation δn+1 C n δn P an ≤ implies that δn = O(δ1 ) for a < 2, which a posteriori justifies the as- 2 sumption of smallness of n δn. It is easy to conclude by propagating bounds inductively.  P 8.3. Large time estimates Now the goal is to go for long-time estimates on the layers hn, and establish Proposition 8.4. Let f 0 = f 0(v) be a spatially homogeneous pro- 0 −2πλ0|η| 0 file satisfying f (η) = O(e ) and f λ ;1 < + , together with | | k kZ 0 ∞ the generalized Penrose stability condition with stability width λL > 0. Let W be an interactione potential with W = O(1/ k 2). Let λ > 0, ′ | | λ < min(λ,λ0,λL), µ> 0, a (1, 2); then there is ε∗ > 0 such that if 0 ∈ fi satisfies fi f Zλ,µ;1 ε ε∗, then c k − k ≤ ≤ n an n N, t 0 h (t) λ,µ;1 Cε . ∀ ∈ ∀ ≥ k kZt ≤ This is much more tricky than the short-time estimates. In par- ticular, it will involve a Lagrangian point of view, where we will use trajectories induced by the force field rather than just free transport; and it will use the estimates from Chapters 6 and 7. The same global strategy of stratification of estimates will be useful, but a number of auxiliary estimates will be propagated. I shall only describe the main ideas in a sketchy way. Sketch of proof of Proposition 8.4. First the general pic- ture: Starting from ∂ hn+1 + v hn+1 + F [f n] hn+1 + F [hn+1] f n = F [hn] hn, t · ∇x · ∇v · ∇v − · ∇v 8.3. LARGE TIME ESTIMATES 105

n n+1 we formally get rid of the term F [f ] vh by using the characteris- tics Sn associated with the force field·∇F [f n](t, x). This gives a kind of reaction equation in the style of (7.1), except that everything is com- posed with Sn. A notable unpleasant consequence is that we lose the n n gradient property: ( vf ) S is not a gradient any longer; as a conse- quence, an additional∇ zero◦ mode will appear in the reaction estimates, associated with an instantaneous response. Fortunately, this term will be uniformly bounded in time. The source term, quadratic in hn, will not cause any serious problem. From there, one sets up a constructive loop in the estimates: If the characteristics are close to the free transport trajectories, then the flow will have good mixing properties, and as a consequence the density will be uniformly smooth and the force will damp to 0. Conversely, if the force damps, then the characteristics remain close to free transport trajectories. To quantify how close the characteristics are from free transport, one introduces the deflection operators (8.12) Ω = Sn S0 , t,τ t,τ ◦ τ,t where Sn is the flow generated by the force F [hn]. Now the core of the proof is to estimate simultaneously hn Ωn−1 (the density hn along the characteristics generated by f n−1) and◦ ρn = hn dv (the spatial density). The density ρn is estimated in the natural gliding regularity: that is, in the space λnτ+µn at time τ. But the composedR density hn Ωn−1 is estimatedF with a twist on the indices, depending on the final◦ time t: using (5.14), the time velocity and regularity indices will be modulated by a function b(t)= B/(1 + t) So the main estimates to propagate are

n n n−1 λ τ+µ (8.13) sup ρ (τ) F n n δn, sup hτ Ωt,τ Zλn(1+b),µn;1 δn, τ− bt τ≥0 k k ≤ t≥τ≥0 ◦ 1+b ≤ where δn is a sequence of positive numbers which should converge very fast to 0. I shall write δ = δ1: this is an estimate of the final error. A number of auxiliary estimates will be propagated. Writing Ω = Ωt,τ and h = h(τ) for simplicity, the desired estimates are (in appro- priate norms) Ωn Id and Ωn I are O(δ/τ s), uniformly in n; • − ∇ − n k k −1 n s Ω Ω and (Ω ) Ω Id are O(δk/τ ), for all k n (so these• expressions− are small as◦ k − , uniformly in n); ≤ →∞ 106 8. NEWTON’S SCHEME

k n−1 k n−1 k n−1 h Ω , xh Ω , ( v + τ x)h Ω are all O(δk), for all k• n;◦ ∇ ◦ ∇ ∇ ◦ ≤ 2hk Ωn−1 is O(τ 2 δ ), for all k n; • ∇ ◦ k ≤ ( hn) Ωn−1 (hn Ωn−1) is O(δ /τ s), for all n. • ∇ ◦ − ∇ ◦ n In the above expressions s> 0 is as large as desired; for the sequel it would be sufficient to choose s = 4, but in these and related estimates I shall continue to write s, meaning an integer which can change from line to line and can be fixed arbitrarily large in advance. The possibility of choosing s large comes from the fact that the deflection estimates of Chapter 6 are exponentially small in τ. Let us see in a sketchy way how this works. To simplify notation, I shall forget about the x-regularity and the parameter µ in the estimate of the kinetic distribution, and focus on the v-regularity parameter λ. At each stage of the iteration a bit or regularity is lost in v (when I say a bit, this is still an infinite number of derivatives, but in the parameter 2 λ of analytic regularity this is just a bit): say λn λn+1 =Λ/n with Λ very small. At each stage there are a number of− steps and at each step a little bit is lost, so five or six intermediate indices are used between λn and λ , say λ >λ′ >λ′′ >...>λ ; but λ λ′ , λ′′ λ′ , etc. are n+1 n n n n+1 n − n n − n all of the order λn λn+1, so I will just forget about the intermediate indices and use the− fact that these differences are all of order O(1/n2). Also I shall assume that δk decreases very fast to 0: in fact this can only be checked a posteriori once all estimates have been performed. With these conventions in mind, let us sketch the steps of the iter- ation.

(a) In this step one shows that Ωn is asymptotically close to Id . n From ρ F λnτ = O(δn) we deduce, as in Chapter 6, a bound on n k k Ωt,τ Id . If we do it in gliding regularity about λn+1, then to ap- − 1/3 ply Proposition 6.1, we need λn λn+1 to be at least of order δ , because the force field is of size δ−(it depends not only on hn but also on h1, h2, . . . ). Of course we cannot afford to lose such a fixed amount of regularity as n . So we modify the estimates in Proposition 6.1 →∞ tby letting the velocity regularity depend on τ and t: replace λn by Zτ λn(1+b) , where b(t)= B/(1 + t). This works because Zτ−bt/(1+b) F n(t) decays faster than λ b(t), • k k n bt λn(1 + b) τ 1+b < λn τ, at least for t positive enough. So • − λ (1+b) we can still estimate the n norm of the force field by its λnτ Z
τ−bt/(1+b) F norm (recall (5.14)). 8.3. LARGE TIME ESTIMATES 107

This does not work for small values of t, but short times have al- ready been treated separately, as explained in Section 8.2. In all the rest of the argument one should consider separately small and not so small times. Then, by amplification of the fixed point technique of Chapter 6, we arrive at

n δ (8.14) Ωt,τ Id Zλn(1+b) = O s . k − k τ− bt τ 1+b   Next, with another fixed point argument, one establishes

n δ (8.15) Ωt,τ I Zλn(1+b) = O s . k∇ − k τ− bt τ 1+b  

(b) In this step one shows that Ωk is close to Ωn, uniformly in n k. This is done again by fixed point, and one arrives at something like≥

n k δk (8.16) Ωt,τ Ωt,τ Zλn(1+b) = O s . k − k τ− bt τ 1+b   (The regularity is the worst of λn and λk, that is λn; and the size is the worst of δn and δk, that is δk.) The important point is that this estimate goes to 0 as k , uniformly in n. From (8.16) it is deduced (by fixed point again...)→∞

k −1 n δk (8.17) Ωt,τ ) Ωt,τ Id = O . ◦ − Zλn(1+b) τ s τ− bt 1+b   (Inversion with a norm of time index τ is possible only if Ωn Ωk is much smaller than 1/τ; but this is guaranteed by (8.16).) − In the sequel I shall not always write the indices of the norms. (c) The next step is to update the controls on the previous layers hk by taking into account the change of the characteristics; so one should estimate hk Ωn for all k. This is done by composition: ◦ hk Ωn =(hk Ωk−1) (Ωk−1)−1 Ωn , ◦ ◦ ◦ ◦ k k−1 so, as a consequence of the induction assumption (h
Ω )= O(δk) and (8.17), we have ∇ ◦

hk Ωn hk Ωk−1 (hk Ωk−1) Ωk−1 −1 Ωn I ◦ − ◦ ≤ k∇ ◦ k t,τ ◦ t,τ − δk
C δ . ≤ k τ s   108 8. NEWTON’S SCHEME

It follows

k n (8.18) hτ Ωt,τ Zλn(1+b) = O(δk). τ− bt k ◦ k 1+b

k n k n Similar bounds are established for ( xh ) Ω and (( v +τ x)h ) Ω , ∇ ◦ ∇ ∇ k ◦ n with just a small loss on the regularity index. Consequently ( vh ) Ω is O(τ δ ). Similarly, ( 2hk) Ωn = O(τ 2 δ ). ∇ ◦ k ∇ ◦ k (d) Next is the key step where an estimate is obtained for ρn+1. First write the equation for hn+1, then the method of characteristics in the force field F [f n] yields

t ρn+1(t, x)= F [hn+1] f n τ,Sn (x, v) dv dτ − · ∇v t,τ Z0 Z + quadratic
contribution
from hn. To take advantage of the mixing effect of the free transport semigroup, introduce the deflection Ωn by force, rewriting the estimate above as

(8.19) t ρn+1(t, x)= F [hn+1] f n Ωn x v(t τ), v dv dτ − · ∇v ◦ t,τ − − Z0 Z h
i
r 2 + O(n δn), where the contribution from hn has been estimated in a crude way. If it was not for the composition by Ωn, we would be in the same situation as in Chapter 7, and we could use the estimates on the Vlasov equation seen as a reaction equation. But (F [hn+1] f n) Ωn does not ·∇v ◦ have the structure G(t, x) vg(t, x, v) which was crucial in Chapter 7. The problem is to show that· ∇ composition by Ωn does not change much in the long run. So we decompose that reaction term as follows: (8.20) F [hn+1] f n Ωn =F [hn+1] f 0 + hk Ωk−1 · ∇v ◦ t,τ · ∇v ◦ k≤n
 X  + F [hn+1] Ωn F [hn+1] ( f n Ωn) ◦ − · ∇v ◦   + F [hn+1] ( hk) Ωk−1 hk Ωk−1 · ∇v ◦ − ∇v ◦ k≤n Xh
i + F [hn+1] ( hk) Ωn ( hk) Ωk−1 . · ∇v ◦ − ∇v ◦ Xk≤nh i 8.3. LARGE TIME ESTIMATES 109

The first term on the right-hand side is fine, and the other three terms will be treated as perturbations in large time, as follows. The second term in the right-hand side of (8.20) is estimated as follows:•

F [hn+1] Ωn F [hn+1] f n Ωn ◦ − k∇v ◦ k n+1 n n n F [h ] Ω Id vf Ω ≤ ∇ k − k k∇ ◦ k δτ C ρn+1 , ≤ k k τ s where I used F [hn+1] C ρn+1 : indeed, thanks to the assump- k∇ k ≤ k k tion W (k) = O(1/ k 2), passing from the density to the force should gain| at least| one derivative.| | (Note carefully: Here we cannot afford to lose regularityc on ρn+1 because we are trying to get a Gronwall-type estimate on the unknown ρn+1, so it is crucial to use the very same norm on the left-hand side and the right-hand side, and the only thing we can use to regain the derivative is the smoothing induced by the convolution inside the force.) Next, by recursion hypothesis, • δ (hk Ωk−1) ( hk) Ωk−1 = O k , ∇v ◦ − ∇v ◦ τ s   which allows to control the third term in the right-hand side of (8.20) by ρn+1 δ /τ s. k k k Finally, to handle the last term in the right-hand side of (8.20), one• writes

( hk) Ωk−1 ( hk) Ωn ∇v ◦ − ∇v ◦

sup 2hk (1 θ) Ωk−1 + θ Ωn Ωk−1 Ωn . ≤ 0≤θ≤1 ∇ ◦ − k − k

From (8.17) the argument of 2hk is close to Ωk−1, uniformly in θ, so up to a slight loss we end up∇ with a bound like

δ 2hk Ωk−1 Ωk−1 Ωn C δ τ 2 k , ∇ ◦ k − k ≤ k τ s   after use of the induction hypothesis on 2hk Ωk−1 and the bound (8.16). ∇ ◦ 110 8. NEWTON’S SCHEME

Plugging all these controls in (8.20) shows

t n+1 n+1 0 k k−1 ρ (t, x)= F [hτ ] v f + hτ Ωt,τ (x v(t τ), v) dτ dv − 0 ·∇ ◦ − − Z  Xk≤n  t ρn+1(τ) + O k k dτ + O(nrδ2). 1+ τ s n Z0  Then one can operate as in Chapter 7 and get a Gronwall estimate n+1 on ρ (τ) F λn+1τ . The difference is an additional term in the kernel K(t,k τ), whichk is O(δ τ −s), uniformly in t. But this is harmless: think indeed that a solution of t ϕ(τ) ϕ(t) A + δ dτ ≤ 1+ τ 2 Z0 satisfies ϕ(t) A + C δ. The robustness of the moment estimates from Chapter 7 and≤ the L2 method in Lemma 3.5 makes it possible to adapt all these estimates to the present complicated situation, yielding in the end n+1 nK r 2 ρ λ τ = O e n δ . k kF n+1 n So this step gives  

n+1 nK r 2 (8.21) ρ λ τ = O(δ ), δ = O e n δ . k kF n+1 n+1 n+1 n 2   n 2n This is not as good as δn+1 = O(δn), and does not imply δn = O(C δ ) as in the classical Newton scheme; but this is still compatible with an δn = O(δ ), a< 2. (e) From the estimate on ρn+1 we immediately deduce an estimate n+1 on the force: F [h ] λ τ = O(δ ). k kF n+1 n+1 (f) Then use the equation for hn+1 once more, but now compose it with Ωn where τ is given, and estimate hn+1 Ωn . This is not so t,τ τ ◦ t,τ difficult as Step (d) because now there is no need to use the same norm on both sides: we already have an estimate on the force, we don’t need any Gronwall-type inequality, we can afford to lose a little bit on the regularity of hn+1 compared with the regularity of ρn+1. After some computations, one gets something like

n+1 n (8.22) hτ Ωt,τ = O(δn+1). ◦ Zλn(1+b) τ− bt 1+b

(g) Deduce (by gliding regularity) that (8.23) (hn+1 Ωn )= O(δ τ), 2(hn+1 Ωn )= O(δ τ 2). ∇ τ ◦ t,τ n+1 ∇ τ ◦ t,τ n+1 8.4. MAIN RESULT 111

(h) Deduce that ( hn+1) Ωn =( Ωn)−1 (hn+1 Ωn)= O(δ τ), ∇ ◦ ∇ ∇ ◦ n+1 and similarly 2hn+1 Ωn = O(δ τ 2). Finally, note that ∇ ◦ n+1 (hn+1 Ωn) ( hn+1) Ωn (Ωn Id ) hn+1 Ωn ∇ ◦ − ∇ ◦ ≤ ∇ − ∇ ◦ δ C (δ τ), ≤ τ s n+1   so at the same time this is small like O(δn+1), and it decays fast in τ. Once this is done, all the estimates have been propagated from stage n to stage n +1, and we can go on!  8.4. Main result Once we have obtained the estimates on all hn, it is easy to conclude the proof of Theorem 4.1. Let us sketch the argument. Summing the estimates on all hn, one obtains the uniform bound 0 0 (8.24) sup f(t, ) f Zλ,µ;1 + sup ρ ρ F λt+µ = O(δ). t≥0 k · − k t t≥0 k − k This bound is the true main result: actually, it contains much more information than Theorem 4.1. It implies that the force F (t, ) satis- · fies F (t) F λt+µ = O(δ), and since F is a gradient, the latter bound impliesk thatk F (t) decays exponentially fast with t. On the other hand, vf grows at most linearly in t, so F vf decays exponentially fast ∇in gliding regularity. This implies that· ( ∇d/dt)f(t, x + vt, v) also decays exponentially; in particular, f(t, x+vt, v) has a large-time limit g(x, v), which is analytic, and the convergence actually holds in an appropriate function space. As a consequence, f(t, x, v) has the same asymp- toticZ behavior as g(x vt, v), which converges weakly to g (v). The conclusion of Theorem− 4.1 follows easily. h i I shall conclude with a few indications on non-analytic data (Theo- rem 4.3). It was noted in Chapter 7 that the expected loss of regularity is like a fractional exponential, say e|ξ|α . So it is expected that all re- sults hold true in Gevrey-ν regularity for any ν > 1/α. All the large-time estimates can indeed be adapted to this setting, either by changing all our norms to handle Gevrey regularity, or by decomposing a Gevrey function into a sum of analytic contributions with analyticity width going to 0 in a controlled way. In practice, we 0 n decompose the initial datum fi f in a sum of data hi , such that n − ′ hi satisfies some analyticity condition in a strip of width λn, and the 112 8. NEWTON’S SCHEME

n n norm of hi decays in a controlled way as n . Then we use hi as an initial datum in the step n of the Newton→ scheme.∞ Of course our n ′ large time estimates on h only hold in regularity less than λn, but the ′ way λn goes to 0 is controlled, so in the end we can reconstruct Gevrey regularity for hn, losing just a bit on the Gevrey exponent in the process. Then the iteration can be performed as in the analytic case. P

Bibliographical notes Newton presented his approximation scheme in a 1669 treatise [82] which was published only decades later. The presentation and the study of the scheme were revised and improved by a series of English mathematicians: Wallis, Raphson, Simpson, Cayley. An ancestor of the Newton scheme is the so-called Babylonian method for the numerical solution of square roots, which some experts conjecture to have been known to Babylonian mathematicians as early as 1900 BC, and to Indian mathematicians before 800 BC: to compute e.g. √2 apply the Newton scheme to the function Φ(x)= x2 2. Kolmogorov’s perturbation theorem for− Hamiltonian systems was announced in [56] in analytic regularity, and Nash’s embedding the- orem appeared in [80]. Kolmogorov’s sketchy proof did not convince everybody at the time, which was very fortunate since it motivated Moser to devise his own proof [76, 77] in a differentiable setting, using Nash’s work as an inspiration. Around the same time, Kolmogorov’s analytic result was, after all, validated by Arnold [5] with an alterna- tive proof. Much later, Chierchia [27] reconstructed the details of was is likely to have been Kolmogorov’s original argument. Chierchia wrote a survey of KAM theory for the online encyclopedia Scholarpedia [28]. The Nash–Moser technique is the main subject of [3]. For the proof of the core Nash–Moser theorem, one may consult [94]. A fixed point approach to the KAM theory was proposed by Herman [47]; while it does not seem to apply in full generality, it does suffice to cover certain simple situations. A fixed point approach to Nash’s embedding theorem was devised by G¨unther [41]. The Cauchy–Kowalevsakaya method is presented in a number of sources; Nirenberg’s presentation [83] is based on a Newton scheme, and is close in spirit to the treatment sketched in these notes, whose details are provided in [78]. (I learnt about Nirenberg’s work from Klainerman, Alinhac and G´erard after [78] was written.) Once again, after a few years, Nirenberg’s use of a Newton scheme has been replaced by a fixed-point theorem [84], and maybe this will also happen some day for our theory of Landau damping. BIBLIOGRAPHICAL NOTES 113

Short-time analyticity estimates on the solutions of the Vlasov– Poisson equation go back to Benachour [13], with an alternative method. A few remarks about Lemma 8.3 can be made. Differentiation of the norm with respect to time-dependent integrability index is classical in the field of hypercontractivity [40]. Differentiation with respect to a time-dependent regularity index is not so common, but appears in the work of Chemin [26] on the short-time regularity of the incompressible Navier–Stokes system. The long-time analysis of the Newton scheme is performed in painful detail in [78]. The adaptation to Gevrey data is sketched in the same source. As I learnt later, Moser already used the idea to decompose a smooth, nonanalytic function h into a sum of analytic functions hn whose norm and analyticity width decay in a controlled way as n . →∞

CHAPTER 9

Conclusions

The main result in this course is that Landau damping survives nonlinearity, and the long-time behavior of the linearized Vlasov equa- tion is, after all, a good approximation of the long-time behavior of the nonlinear Vlasov equation. This ends up a controversy and pro- vides an answer to the objection formulated by Backus half a century ago. In the end Landau was right, although the proof involves many ingredients which were inaccessible at his time. Remarkably, the range of interactions which are admissible in the main result includes the Poisson coupling (repulsive or attractive) as a limit case. Moreover, the theory provides an interpretation of Landau damp- ing: this is a relaxation by mixing, confinement and smoothness. The mixing transport equation converts smoothness into decay, in the spirit of Fourier transform (Riemann–Lebesgue lemma). Regular- ity goes away from the v variable to the x variable, so the force becomes very smooth, and because it has a gradient structure this implies time decay. A global stability condition has to be imposed on the linearized problem; this is what I called the generalized Landau–Penrose, or just generalized Penrose condition. To handle this leak of regularity, the notion of gliding regularity was presented in these notes: the regularity is quantified by comparison to the solution of the free transport equation, at some time. Even though the solution of the linearized problem involves a loss of gliding regularity (which implies relaxation), regularity estimates survive the nonlinear perturbation by a mathematical (rather than physical) phenomenon comparable to the KAM theory, which takes advantage of the complete integrability of the original system (in our case the linearized Vlasov equation) and a Newton scheme to overcome the loss of regularity. In this sense the proof provides an unexpected bridge between three of the most famous paradoxical statements from classical mechanics in the twentieth century: Landau damping, KAM theory, and the echo

115 116 9.CONCLUSIONS experiment. This is all the more remarkable that this bridge only ap- pears in the treatment of the nonlinear Vlasov equation, while Landau was dealing specifically with the linearized equation. The fully constructive property of the Newton scheme allows to construct the asymptotic state, opening the door to asymptotic studies. For instance, one can construct in this way heteroclinic trajectories of the nonlinear Vlasov equation (solutions are automatically homoclinic at order 2 in the perturbation size ε; but heteroclinic corrections of order O(ε3) can appear). This shows that the asymptotic behavior cannot be predicted on the basis on invariants of motion alone: indeed, these invariants are all preserved by the reversal of velocities, which amounts to a change of the direction of time. Another striking feature of our proof is that, compared to KAM theory, the loss of regularity is much more severe in the present case: infinitely many derivatives are lost, corresponding to a fractional ex- ponential in Fourier space. Such high losses prevent the application of the classical Nash–Moser regularization scheme; for this reason in par- ticular, we have not been able to establish nonlinear Landau damping below a certain Gevrey regularity. The issue of nonlinear Landau damping for less smooth data ap- pears wide open. In a recent contribution, Lin proved that one cannot hope for Landau damping in low regularity, that is, with less than 2 derivatives in an appropriate Sobolev space. Indeed, in such a low regularity topology, BGK waves are dense around stable homogeneous equilibrium profiles; so damping to a homogeneous state might still be true for typical solutions, but cannot hold over a whole neighborhood of the equilibrium. Another development of interest would be the adaptation of Lan- dau damping theory to other models sharing some similar features. For the most natural candidate, the two-dimensional incompressible Euler equation, this turns out to be much more difficult than could be ex- pected; one reason for this is that the damping rate is so slow that the deflection estimates are terrible. Yet another interesting issue would be to understand Landau damp- ing when the geometry of the asymptotic flow in phase space is more complicated than the periodic shear flow geometry induced by free transport. Since in principle the main ingredients are confinement and phase mixing, one may expect Landau damping to hold in a general situation where these two properties are satisfied, and it would be easy to cook up such model equations. But then the so handy Fourier trans- form cannot be used, so the proof may be much more complicated. BIBLIOGRAPHICAL NOTES 117

The benchmarking of reliable long-time numerical schemes, the study of the linear and nonlinear stability of BGK waves, the qualita- tive study of large perturbations of equilibrium, the statistical theory of the Vlasov–Poisson equation, remain wide open fascinating subjects. Bibliographical notes KAM type problems with a loss of infinitely many derivatives (mul- tiplication by a fractional exponential in Fourier space) have been con- sidered by Popov [88]; in this case (as I learnt from Chierchia and P¨oschel) nobody knows how to treat Cr regularity in the style of Moser [76]. Heteroclinic solutions of the nonlinear Vlasov equation are con- structed in [78, Section 14]. Lin’s negative results are presented in [63]. The precise statement is that there is density in W 1+1/p,p+0 topology for any p (1, ). A preliminary discussion of nonlinear Landau damping∈ ∞ for two- dimensional incompressible Euler equation was performed by Bouchet and Morita [17]. Together with Mouhot, we tried to put this on rig- orous footing by adapting the study of the nonlinear Vlasov equation, but stumbled upon formidable difficulties, even in the simple case of a perturbation of a linear shear flow [79].

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